NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

Q: Examine the following functions for continuity: (a) $f(x)=x-5$ (b) $f(x)=\dfrac1{x-5}, x\ne5$ (c) $f(x)=\dfrac{x^2-25}{x+5}, x\ne-5$ (d) $f(x)=|x-5|$.

(a) Continuous on $\mathbb{R}$. (b) Continuous for $x\ne5$ and discontinuous at $x=5$. (c) Continuous for $x\ne-5$ and discontinuous at $x=-5$. (d) Continuous on $\mathbb{R}$.

Q: Is the function $f(x)=\begin{cases}x, & x\le1\\5, & x\gt1\end{cases}$ continuous at $x=0$? At $x=1$? At $x=2$?

Continuous at $x=0$ and $x=2$; discontinuous at $x=1$.

Q: Is $f(x)=\begin{cases}x+5, & x\le1\\x-5, & x\gt1\end{cases}$ a continuous function?

No; it is discontinuous at $x=1$.

Q: Discuss the continuity of $f(x)=\begin{cases}3, & 0\le x\le1\\4, & 1\lt x\lt3\\5, & 3\le x\le10\end{cases}$.

Discontinuous at $x=1$ and $x=3$; continuous elsewhere in its domain.

Q: For what value of $\lambda$ is $f(x)=\begin{cases}\lambda(x^2-2x), & x\le0\\4x+1, & x\gt0\end{cases}$ continuous at $x=0$? What about continuity at $x=1$?

No $\lambda$ works at $x=0$; it is continuous at $x=1$ for every $\lambda$.

Your Progress - Chapter 50% complete

1Exercise 5.134 questions

Q.1Prove that the function

$f(x)=5x-3$

is continuous at

$x=0$

, at

$x=-3$

and at

$x=5$

Solution

As a polynomial, $$5x-3$$ is continuous for every real $$x$$ , so $\lim_{x\to a}f(x)=f(a)$ for $$a=0,-3,5$$ .

Answer:

Continuous at all three points.

Q.2Examine the continuity of the function

$f(x)=2x^2-1$

$x=3$

Solution

$\lim_{x\to3}(2x^2-1)=17=f(3)$ , so the function is continuous at $$3$$ .

Answer:

Continuous at $$x=3$$ .

Q.3Examine the following functions for continuity: (a)

$f(x)=x-5$

(b)

f(x)=\dfrac1{x-5}, x\ne5

(c)

f(x)=\dfrac{x^2-25}{x+5}, x\ne-5

(d)

$f(x)=|x-5|$

Solution

Polynomials and absolute-value functions are continuous everywhere; rational functions are continuous wherever their denominators are non-zero.

Answer:

(a) Continuous on $\mathbb{R}$ . (b) Continuous for $x\ne5$ and discontinuous at $$x=5$$ . (c) Continuous for $x\ne-5$ and discontinuous at $$x=-5$$ . (d) Continuous on $\mathbb{R}$ .

Q.4Prove that the function

$f(x)=x^n$

is continuous at

$x=n$

, where

$n$

is a positive integer.v

Solution

$$x^n$$ is a polynomial, hence $\lim_{x\to n}x^n=n^n=f(n)$ .

Answer:

Continuous at $$x=n$$ .

Q.5Is the function

f(x)=\begin{cases}x, & x\le1\\5, & x\gt1\end{cases}

continuous at

$x=0$

? At

$x=1$

? At

$x=2$

Solution

Near $$0$$ , $$f=x$$ ; near $$2$$ , $$f=5$$ . At $$1$$ , $$f(1)=1$$ but the right-hand limit is $$5$$ .

Answer:

Continuous at $$x=0$$ and $$x=2$$ ; discontinuous at $$x=1$$ .

Q.6Find all points of discontinuity of

f(x)=\begin{cases}2x+3, & x\le2\\2x-3, & x\gt2\end{cases}

Solution

At $$2$$ , the left value is $$7$$ and the right-hand limit is $$1$$ ; elsewhere each piece is polynomial.

Answer:

Only $$x=2$$ .

Q.7Find all points of discontinuity of

f(x)=\begin{cases}|x|+3, & x\le-3\\-2x, & -3\lt x\lt3\\6x+2, & x\ge3\end{cases}

Solution

At $$-3$$ , both side limits are $$6$$ . At $$3$$ , the left limit is $$-6$$ but $$f(3)=20$$ .

Answer:

Only $$x=3$$ .

Q.8Find all points of discontinuity of

f(x)=\begin{cases}\dfrac{|x|}{x}, & x\ne0\\0, & x=0\end{cases}

Solution

For $x\lt0$ , $$|x|/x=-1$$ ; for $x\gt0$ , $$|x|/x=1$$ , so the one-sided limits at $$0$$ differ.

Answer:

Only $$x=0$$ .

Q.9Find all points of discontinuity of

f(x)=\begin{cases}\dfrac{x}{|x|}, & x\lt0\\-1, & x\ge0\end{cases}

Solution

For $x\lt0$ , $$x/|x|=-1$$ , and for $x\ge0$ the function is also $$-1$$ .

Answer:

No points of discontinuity.

Q.10Find all points of discontinuity of

f(x)=\begin{cases}x+1, & x\ge1\\x^2+1, & x\lt1\end{cases}

Solution

At $$x=1$$ , the left limit is $$2$$ and $$f(1)=2$$ ; elsewhere the pieces are polynomials.

Answer:

No points of discontinuity.

Q.11Find all points of discontinuity of

f(x)=\begin{cases}x^3-3, & x\le2\\x^2+1, & x\gt2\end{cases}

Solution

At $$x=2$$ , both side limits equal $$5$$ and $$f(2)=5$$ .

Answer:

No points of discontinuity.

Q.12Find all points of discontinuity of

f(x)=\begin{cases}x^{10}-1, & x\le1\\x^2, & x\gt1\end{cases}

Solution

At $$1$$ , the left value is $$0$$ while the right-hand limit is $$1$$ .

Answer:

Only $$x=1$$ .

Q.13Is

f(x)=\begin{cases}x+5, & x\le1\\x-5, & x\gt1\end{cases}

a continuous function?v

Solution

At $$1$$ , $$f(1)=6$$ but the right-hand limit is $$-4$$ .

Answer:

No; it is discontinuous at $$x=1$$ .

Q.14Discuss the continuity of

f(x)=\begin{cases}3, & 0\le x\le1\\4, & 1\lt x\lt3\\5, & 3\le x\le10\end{cases}

Solution

The jumps are from $$3$$ to $$4$$ at $$1$$ and from $$4$$ to $$5$$ at $$3$$ .

Answer:

Discontinuous at $$x=1$$ and $$x=3$$ ; continuous elsewhere in its domain.

Q.15Discuss the continuity of

f(x)=\begin{cases}2x, & x\lt0\\0, & 0\le x\le1\\4x, & x\gt1\end{cases}

Solution

At $$0$$ , both side limits equal $$0$$ . At $$1$$ , the left limit is $$0$$ and the right-hand limit is $$4$$ .

Answer:

Discontinuous only at $$x=1$$ .

Q.16Discuss the continuity of

f(x)=\begin{cases}-2, & x\le-1\\2x, & -1\lt x\le1\\2, & x\gt1\end{cases}

Solution

At $$-1$$ the joining value is $$-2$$ , and at $$1$$ the joining value is $$2$$ .

Answer:

Continuous for all real $$x$$ .

Q.17Find the relationship between

$a$

and

$b$

so that

f(x)=\begin{cases}ax+1, & x\le3\\bx+3, & x\gt3\end{cases}

is continuous at

$x=3$

Solution

Continuity gives $$3a+1=3b+3$$ , so $$3(a-b)=2$$ .

Answer:

$a-b=\dfrac23$ .

Q.18For what value of

\lambda

f(x)=\begin{cases}\lambda(x^2-2x), & x\le0\\4x+1, & x\gt0\end{cases}

continuous at

$x=0$

? What about continuity at

$x=1$

Solution

At $$0$$ , $$f(0)=0$$ but the right-hand limit is $$1$$ . At $$1$$ , the formula $$4x+1$$ applies locally.

Answer:

No $\lambda$ works at $$x=0$$ ; it is continuous at $$x=1$$ for every $\lambda$ .

Q.19Show that

$g(x)=x-[x]$

is discontinuous at all integral points, where

$[x]$

is the greatest integer less than or equal to

$x$

Solution

At an integer $$n$$ , $$g(n)=0$$ but $\lim_{x\to n^-}(x-[x])=1$ .

Answer:

Discontinuous at every integer.

Q.20Is

f(x)=x^2-\sin x+5

continuous at

x=\pi

Solution

$$x^2$$ , $\sin x$ and constants are continuous, so their combination is continuous at $\pi$ .

Answer:

Yes.

Q.21Discuss the continuity of (a)

\sin x+\cos x

(b)

\sin x-\cos x

(c)

\sin x\cos x

Solution

Sums, differences and products of the continuous sine and cosine functions are continuous.

Answer:

All three are continuous on $\mathbb{R}$ .

Q.22Discuss the continuity of the cosine, cosecant, secant and cotangent functions.v

Solution

The reciprocal/quotient trigonometric functions are continuous wherever their denominators are non-zero.

Answer:

$\cos x$ is continuous on $\mathbb{R}$ ; $\csc x$ and $\cot x$ for $x\ne n\pi$ ; $\sec x$ for $x\ne\dfrac\pi2+n\pi$ .

Q.23Find all points of discontinuity of

f(x)=\begin{cases}\dfrac{\sin x}{x}, & x\lt0\\x+1, & x\ge0\end{cases}

Solution

At $$0$$ , $\lim_{x\to0^-}\sin x/x=1=f(0)$ ; elsewhere the pieces are continuous.

Answer:

No points of discontinuity.

Q.24Determine if

f(x)=\begin{cases}x^2\sin\dfrac1x, & x\ne0\\0, & x=0\end{cases}

is a continuous function.v

Solution

At $$0$$ , $|x^2\sin(1/x)|\le x^2\to0=f(0)$ ; elsewhere it is continuous.

Answer:

Yes, continuous on $\mathbb{R}$ .

Q.25Examine the continuity of

f(x)=\begin{cases}\sin x-\cos x, & x\ne0\\-1, & x=0\end{cases}

Solution

At $$0$$ , $\lim_{x\to0}(\sin x-\cos x)=-1=f(0)$ .

Answer:

Continuous on $\mathbb{R}$ .

Q.26Find

$k$

so that

f(x)=\begin{cases}\dfrac{k\cos x}{\pi-2x}, & x\ne\dfrac\pi2\\3, & x=\dfrac\pi2\end{cases}

is continuous at

x=\dfrac\pi2

Solution

$\lim_{x\to\pi/2}\cos x/(\pi-2x)=1/2$ , so $$k/2=3$$ .

Answer:

$$k=6$$ .

Q.27Find

$k$

so that

f(x)=\begin{cases}kx^2, & x\le2\\3, & x\gt2\end{cases}

is continuous at

$x=2$

Solution

Continuity gives $$4k=3$$ .

Answer:

$k=\dfrac34$ .

Q.28Find

$k$

so that

f(x)=\begin{cases}kx+1, & x\le\pi\\\cos x, & x\gt\pi\end{cases}

is continuous at

x=\pi

Solution

Continuity gives $k\pi+1=\cos\pi=-1$ .

Answer:

$k=-\dfrac2\pi$ .

Q.29Find

$k$

so that

f(x)=\begin{cases}kx+1, & x\le5\\3x-5, & x\gt5\end{cases}

is continuous at

$x=5$

Solution

Continuity gives $$5k+1=10$$ .

Answer:

$k=\dfrac95$ .

Q.30Find

$a$

and

$b$

such that

f(x)=\begin{cases}5, & x\le2\\ax+b, & 2\lt x\lt10\\21, & x\ge10\end{cases}

is continuous.v

Solution

Continuity gives $$2a+b=5$$ and $$10a+b=21$$ ; solving gives $$a=2,b=1$$ .

Answer:

$$a=2$$ , $$b=1$$ .

Q.31Show that

f(x)=\cos(x^2)

is continuous.v

Solution

It is the composition of continuous functions $$x^2$$ and $\cos x$ .

Answer:

Continuous on $\mathbb{R}$ .

Q.32Show that

f(x)=|\cos x|

is continuous.v

Solution

It is the composition of the continuous functions $\cos x$ and $$|x|$$ .

Answer:

Continuous on $\mathbb{R}$ .

Q.33Examine that

\sin|x|

is a continuous function.v

Solution

It is the composition of the continuous functions $$|x|$$ and $\sin x$ .

Answer:

Continuous on $\mathbb{R}$ .

Q.34Find all points of discontinuity of

$f(x)=|x|-|x+1|$

Solution

Both absolute-value terms are continuous, so their difference is continuous.

Answer:

No points of discontinuity.

2Exercise 5.210 questions

Q.1Differentiate

\sin(x^2+5)

with respect to

$x$

Solution

Use the chain rule with inner function $$x^2+5$$ .

Answer:

$2x\cos(x^2+5)$ .

Q.2Differentiate

\cos(\sin x)

with respect to

$x$

Solution

Use the chain rule: derivative of $\cos u$ is $-\sin u\,u'$ .

Answer:

$-\cos x\sin(\sin x)$ .

Q.3Differentiate

\sin(ax+b)

with respect to

$x$

Solution

Use the chain rule with $$u=ax+b$$ .

Answer:

$a\cos(ax+b)$ .

Q.4Differentiate

\sec(\tan(\sqrt{x}))

with respect to

$x$

Solution

Differentiate the nested functions $\sec u$ , $\tan v$ and $\sqrt{x}$ successively.

Answer:

$\dfrac{\sec(\tan\sqrt{x})\tan(\tan\sqrt{x})\sec^2\sqrt{x}}{2\sqrt{x}}$ .

Q.5Differentiate

\dfrac{\sin(ax+b)}{\cos(cx+d)}

with respect to

$x$

Solution

Apply the quotient rule and the chain rule to numerator and denominator.

Answer:

$\dfrac{a\cos(ax+b)\cos(cx+d)+c\sin(ax+b)\sin(cx+d)}{\cos^2(cx+d)}$ .

Q.6Differentiate

\cos x^3\cdot\sin^2(x^5)

with respect to

$x$

Solution

Use the product rule; differentiate each factor by the chain rule.

Answer:

$-3x^2\sin x^3\sin^2(x^5)+10x^4\cos x^3\sin(x^5)\cos(x^5)$ .

Q.7Differentiate

2\sqrt{\cot(x^2)}

with respect to

$x$

Solution

Write it as $2(\cot x^2)^{1/2}$ and apply the chain rule.

Answer:

$-\dfrac{2x\csc^2(x^2)}{\sqrt{\cot(x^2)}}$ .

Q.8Differentiate

\cos(\sqrt{x})

with respect to

$x$

Solution

Apply the chain rule to $\cos u$ with $u=\sqrt{x}$ .

Answer:

$-\dfrac{\sin\sqrt{x}}{2\sqrt{x}}$ .

Q.9Prove that

$f(x)=|x-1|$

x\in\mathbb{R}

, is not differentiable at

$x=1$

Solution

The left derivative is $$-1$$ and the right derivative is $$1$$ ; they are unequal.

Answer:

Not differentiable at $$x=1$$ .

Q.10Prove that

$f(x)=[x]$

0\lt x\lt3

, is not differentiable at

$x=1$

and

$x=2$

Solution

The greatest integer function has jumps at integers; differentiability would imply continuity.

Answer:

Not differentiable at $$x=1$$ and $$x=2$$ .

3Exercise 5.315 questions

Q.1Find

\dfrac{dy}{dx}

2x+3y=\sin x

Solution

Differentiate implicitly: $2+3y'=\cos x$ .

Answer:

$\dfrac{\cos x-2}{3}$ .

Q.2Find

\dfrac{dy}{dx}

2x+3y=\sin y

Solution

Differentiate implicitly: $2+3y'=\cos y\,y'$ .

Answer:

$\dfrac{2}{\cos y-3}$ .

Q.3Find

\dfrac{dy}{dx}

ax+by^2=\cos y

Solution

Differentiate: $a+2byy'=-\sin y\,y'$ .

Answer:

$-\dfrac{a}{2by+\sin y}$ .

Q.4Find

\dfrac{dy}{dx}

xy+y^2=\tan x+y

Solution

Differentiate and collect terms in $$y'$$ .

Answer:

$\dfrac{\sec^2x-y}{x+2y-1}$ .

Q.5Find

\dfrac{dy}{dx}

$x^2+xy+y^2=100$

Solution

Differentiate: $$2x+y+xy'+2yy'=0$$ .

Answer:

$-\dfrac{2x+y}{x+2y}$ .

Q.6Find

\dfrac{dy}{dx}

$x^3+x^2y+xy^2+y^3=81$

Solution

Implicit differentiation and collection of $$y'$$ gives the result.

Answer:

$-\dfrac{3x^2+2xy+y^2}{x^2+2xy+3y^2}$ .

Q.7Find

\dfrac{dy}{dx}

\sin^2y+\cos xy=\kappa

Solution

Differentiate: $\sin2y\,y'-\sin(xy)(y+xy')=0$ .

Answer:

$\dfrac{y\sin(xy)}{\sin2y-x\sin(xy)}$ .

Q.8Find

\dfrac{dy}{dx}

\sin^2x+\cos^2y=1

Solution

Differentiate: $\sin2x-\sin2y\,y'=0$ .

Answer:

$\dfrac{\sin2x}{\sin2y}$ .

Q.9Find

\dfrac{dy}{dx}

y=\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)

Solution

Use $$u=2x/(1+x^2)$$ and $y'=u'/\sqrt{1-u^2}$ , noting the absolute value in the square root.

Answer:

$\dfrac{2}{1+x^2}$ for $|x|\lt1$ and $-\dfrac{2}{1+x^2}$ for $|x|\gt1$ .

Q.10Find

\dfrac{dy}{dx}

y=\tan^{-1}\left(\dfrac{3x-x^3}{1-3x^2}\right)

-\dfrac1{\sqrt3}\lt x\lt\dfrac1{\sqrt3}

Solution

The argument is $\tan(3\tan^{-1}x)$ on the given interval, so $y=3\tan^{-1}x$ .

Answer:

$\dfrac{3}{1+x^2}$ .

Q.11Find

\dfrac{dy}{dx}

y=\cos^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)

0\lt x\lt1

Solution

Put $x=\tan\theta$ ; then the argument is $\cos2\theta$ and $y=2\tan^{-1}x$ .

Answer:

$\dfrac{2}{1+x^2}$ .

Q.12Find

\dfrac{dy}{dx}

y=\sin^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)

0\lt x\lt1

Solution

With $x=\tan\theta$ , the argument is $\cos2\theta=\sin(\pi/2-2\theta)$ .

Answer:

$-\dfrac{2}{1+x^2}$ .

Q.13Find

\dfrac{dy}{dx}

y=\cos^{-1}\left(\dfrac{2x}{1+x^2}\right)

-1\lt x\lt1

Solution

Put $x=\tan\theta$ ; then $2x/(1+x^2)=\sin2\theta=\cos(\pi/2-2\theta)$ .

Answer:

$-\dfrac{2}{1+x^2}$ .

Q.14Find

\dfrac{dy}{dx}

y=\sin^{-1}\left(2x\sqrt{1-x^2}\right)

-\dfrac1{\sqrt2}\lt x\lt\dfrac1{\sqrt2}

Solution

Put $x=\sin\theta$ ; then the argument is $\sin2\theta$ .

Answer:

$\dfrac{2}{\sqrt{1-x^2}}$ .

Q.15Find

\dfrac{dy}{dx}

y=\sec^{-1}\left(\dfrac1{2x^2-1}\right)

0\lt x\lt\dfrac1{\sqrt2}

Solution

Use $d(\sec^{-1}u)/dx=u'/(|u|\sqrt{u^2-1})$ with $u=(2x^2-1)^{-1}$ .

Answer:

$-\dfrac{2}{\sqrt{1-x^2}}$ .

4Exercise 5.410 questions

Q.1Differentiate

\dfrac{e^x}{\sin x}

w.r.t.

$x$

Solution

Apply the quotient rule.

Answer:

$\dfrac{e^x(\sin x-\cos x)}{\sin^2x}$ .

Q.2Differentiate

e^{\sin^{-1}x}

w.r.t.

$x$

Solution

Use the chain rule.

Answer:

$\dfrac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}$ .

Q.3Differentiate

e^{x^3}

w.r.t.

$x$

Solution

Use the chain rule.

Answer:

$3x^2e^{x^3}$ .

Q.4Differentiate

\sin(\tan^{-1}e^{-x})

w.r.t.

$x$

Solution

Use the chain rule and $\cos(\tan^{-1}u)=1/\sqrt{1+u^2}$ .

Answer:

$-\dfrac{e^{-x}}{(1+e^{-2x})^{3/2}}$ .

Q.5Differentiate

\log(\cos e^x)

w.r.t.

$x$

Solution

Use $d\log u=u'/u$ with $u=\cos e^x$ .

Answer:

$-e^x\tan(e^x)$ .

Q.6Differentiate

e^x+e^{x^2}+\cdots+e^{x^5}

w.r.t.

$x$

Solution

Differentiate each term $e^{x^n}$ .

Answer:

$e^x+2xe^{x^2}+3x^2e^{x^3}+4x^3e^{x^4}+5x^4e^{x^5}$ .

Q.7Differentiate

\sqrt{e^{\sqrt{x}}}

x\gt0

, w.r.t.

$x$

Solution

Write the function as $e^{\sqrt{x}/2}$ and differentiate.

Answer:

$\dfrac{e^{\sqrt{x}/2}}{4\sqrt{x}}$ .

Q.8Differentiate

\log(\log x)

x\gt1

, w.r.t.

$x$

Solution

Use the chain rule.

Answer:

$\dfrac1{x\log x}$ .

Q.9Differentiate

\dfrac{\cos x}{\log x}

x\gt0

, w.r.t.

$x$

Solution

Apply the quotient rule.

Answer:

$\dfrac{-\sin x\log x-\dfrac{\cos x}{x}}{(\log x)^2}$ .

Q.10Differentiate

\cos(\log x+e^x)

x\gt0

, w.r.t.

$x$

Solution

Use the chain rule.

Answer:

$-\sin(\log x+e^x)\left(\dfrac1x+e^x\right)$ .

5Exercise 5.518 questions

Q.1Differentiate

\cos x\cdot\cos2x\cdot\cos3x

w.r.t.

$x$

Solution

Logarithmic differentiation gives $y'/y=-\tan x-2\tan2x-3\tan3x$ .

Answer:

$\cos x\cos2x\cos3x(-\tan x-2\tan2x-3\tan3x)$ .

Q.2Differentiate

\sqrt{\dfrac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}

w.r.t.

$x$

Solution

Take logarithms, differentiate, and multiply by $$y$$ .

Answer:

$\dfrac y2\left(\dfrac1{x-1}+\dfrac1{x-2}-\dfrac1{x-3}-\dfrac1{x-4}-\dfrac1{x-5}\right)$ .

Q.3Differentiate

(\log x)^{\cos x}

w.r.t.

$x$

Solution

Use $d(u^v)=u^v(v'\log u+vu'/u)$ .

Answer:

$(\log x)^{\cos x}\left[-\sin x\log(\log x)+\dfrac{\cos x}{x\log x}\right]$ .

Q.4Differentiate

x^x-2^{\sin x}

w.r.t.

$x$

Solution

Differentiate $$x^x$$ by logarithmic differentiation and $2^{\sin x}$ by the chain rule.

Answer:

$x^x(1+\log x)-2^{\sin x}(\log2)\cos x$ .

Q.5Differentiate

$(x+3)^2(x+4)^3(x+5)^4$

w.r.t.

$x$

Solution

Use logarithmic differentiation.

Answer:

$y\left(\dfrac2{x+3}+\dfrac3{x+4}+\dfrac4{x+5}\right)$ , where $$y$$ is the given product.

Q.6Differentiate

\left(x+\dfrac1x\right)^x+x^{1+1/x}

w.r.t.

$x$

Solution

Apply logarithmic differentiation to both variable-power terms.

Answer:

$\left(x+\dfrac1x\right)^x\left[\log\left(x+\dfrac1x\right)+\dfrac{x(1-x^{-2})}{x+x^{-1}}\right]+x^{1+1/x}\left[\dfrac{1+1/x}{x}-\dfrac{\log x}{x^2}\right]$ .

Q.7Differentiate

(\log x)^x+x^{\log x}

w.r.t.

$x$

Solution

Use logarithmic differentiation on both terms.

Answer:

$(\log x)^x\left(\log(\log x)+\dfrac1{\log x}\right)+x^{\log x}\dfrac{2\log x}{x}$ .

Q.8Differentiate

(\sin x)^x+\sin^{-1}\sqrt{x}

w.r.t.

$x$

Solution

Use logarithmic differentiation and the chain rule.

Answer:

$(\sin x)^x(\log\sin x+x\cot x)+\dfrac1{2\sqrt{x}\sqrt{1-x}}$ .

Q.9Differentiate

x^{\sin x}+(\sin x)^{\cos x}

w.r.t.

$x$

Solution

Use the derivative formula for $$u^v$$ on both terms.

Answer:

$x^{\sin x}\left(\cos x\log x+\dfrac{\sin x}{x}\right)+(\sin x)^{\cos x}\left(-\sin x\log\sin x+\dfrac{\cos^2x}{\sin x}\right)$ .

Q.10Differentiate

x^{x\cos x}+\dfrac{x^2+1}{x^2-1}

w.r.t.

$x$

Solution

Use logarithmic differentiation and the quotient rule.

Answer:

$x^{x\cos x}\{(\cos x-x\sin x)\log x+\cos x\}-\dfrac{4x}{(x^2-1)^2}$ .

Q.11Differentiate

(x\cos x)^x+(x\sin x)^{1/x}

w.r.t.

$x$

Solution

Use logarithmic differentiation separately.

Answer:

$(x\cos x)^x[\log(x\cos x)+1-x\tan x]+(x\sin x)^{1/x}\left[-\dfrac{\log(x\sin x)}{x^2}+\dfrac1x\left(\dfrac1x+\cot x\right)\right]$ .

Q.12Find

\dfrac{dy}{dx}

$x^y+y^x=1$

Solution

Differentiate $$x^y$$ and $$y^x$$ implicitly and solve for $$y'$$ .

Answer:

$-\dfrac{yx^{y-1}+y^x\log y}{x^y\log x+xy^{x-1}}$ .

Q.13Find

\dfrac{dy}{dx}

$y^x=x^y$

Solution

Taking logarithms gives $x\log y=y\log x$ ; differentiate implicitly.

Answer:

$\dfrac{y/x-\log y}{x/y-\log x}$ .

Q.14Find

\dfrac{dy}{dx}

(\cos x)^y=(\cos y)^x

Solution

Take logarithms and differentiate $y\log\cos x=x\log\cos y$ .

Answer:

$\dfrac{\log(\cos y)+y\tan x}{\log(\cos x)+x\tan y}$ .

Q.15Find

\dfrac{dy}{dx}

x^y=e^{x-y}

Solution

Taking logarithms gives $y\log x=x-y$ ; differentiate and solve.

Answer:

$\dfrac{1-y/x}{1+\log x}$ .

Q.16Find the derivative of

$f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)$

and hence find

$f'(1)$

Solution

Logarithmic differentiation gives the formula; at $$x=1$$ , $$f(1)=16$$ and the bracket is $$15/2$$ .

Answer:

$f'(x)=f(x)\left(\dfrac1{1+x}+\dfrac{2x}{1+x^2}+\dfrac{4x^3}{1+x^4}+\dfrac{8x^7}{1+x^8}\right)$ and $$f'(1)=120$$ .

Q.17Differentiate

$(x^2-5x+8)(x^3+7x+9)$

using product rule, expansion and logarithmic differentiation. Do they give the same answer?v

Solution

The product rule gives the expression directly; expansion and logarithmic differentiation simplify to the same derivative.

Answer:

$$(2x-5)(x^3+7x+9)+(x^2-5x+8)(3x^2+7)$$ ; yes, all methods agree.

Q.18If

$u,v,w$

are functions of

$x$

, show that

\dfrac d{dx}(uvw)=\dfrac{du}{dx}vw+u\dfrac{dv}{dx}w+uv\dfrac{dw}{dx}

in two ways.v

Solution

Repeated product rule gives $$(uv)'w+uvw'$$ . Logarithmic differentiation of $$y=uvw$$ gives $$y'/y=u'/u+v'/v+w'/w$$ .

Answer:

$\dfrac d{dx}(uvw)=u'vw+uv'w+uvw'$ .

6Exercise 5.611 questions

Q.1If

$x=2at^2$

$y=at^4$

, find

\dfrac{dy}{dx}

without eliminating the parameter.v

Solution

$$dy/dx=(4at^3)/(4at)=t^2$$ .

Answer:

$$t^2$$ .

Q.2If

x=a\cos\theta

y=b\cos\theta

, find

\dfrac{dy}{dx}

Solution

Divide $dy/d\theta=-b\sin\theta$ by $dx/d\theta=-a\sin\theta$ .

Answer:

$\dfrac ba$ .

Q.3If

x=\sin t

y=\cos2t

, find

\dfrac{dy}{dx}

Solution

$dy/dx=(-2\sin2t)/(\cos t)=-4\sin t$ .

Answer:

$-4\sin t$ .

Q.4If

$x=4t$

y=\dfrac4t

, find

\dfrac{dy}{dx}

Solution

Divide $$dy/dt=-4/t^2$$ by $$dx/dt=4$$ .

Answer:

$-\dfrac1{t^2}$ .

Q.5If

x=\cos\theta-\cos2\theta

y=\sin\theta-\sin2\theta

, find

\dfrac{dy}{dx}

Solution

Differentiate both parametric equations and divide.

Answer:

$\dfrac{\cos\theta-2\cos2\theta}{-\sin\theta+2\sin2\theta}$ .

Q.6If

x=a(\theta-\sin\theta)

y=a(1+\cos\theta)

, find

\dfrac{dy}{dx}

Solution

Divide $dy/d\theta=-a\sin\theta$ by $dx/d\theta=a(1-\cos\theta)$ .

Answer:

$-\dfrac{\sin\theta}{1-\cos\theta}=-\cot\dfrac\theta2$ .

Q.7If

x=\dfrac{\sin^3t}{\sqrt{\cos2t}}

y=\dfrac{\cos^3t}{\sqrt{\cos2t}}

, find

\dfrac{dy}{dx}

Solution

Log-differentiate $$x$$ and $$y$$ , then divide $$dy/dt$$ by $$dx/dt$$ and simplify.

Answer:

$\dfrac{3\tan^2t-1}{\tan t(3-\tan^2t)}$ .

Q.8If

x=a\left(\cos t+\log\tan\dfrac t2\right)

y=a\sin t

, find

\dfrac{dy}{dx}

Solution

$dx/dt=a(-\sin t+\csc t)=a\cos^2t/\sin t$ and $dy/dt=a\cos t$ .

Answer:

$\tan t$ .

Q.9If

x=a\sec\theta

y=b\tan\theta

, find

\dfrac{dy}{dx}

Solution

Divide $b\sec^2\theta$ by $a\sec\theta\tan\theta$ .

Answer:

$\dfrac{b}{a\sin\theta}$ .

Q.10If

x=a(\cos\theta+\theta\sin\theta)

y=a(\sin\theta-\theta\cos\theta)

, find

\dfrac{dy}{dx}

Solution

$dx/d\theta=a\theta\cos\theta$ and $dy/d\theta=a\theta\sin\theta$ .

Answer:

$\tan\theta$ .

Q.11If

x=\sqrt{a^{\sin^{-1}t}}

y=\sqrt{a^{\cos^{-1}t}}

, show that

\dfrac{dy}{dx}=-\dfrac yx

Solution

Write $x=a^{\sin^{-1}t/2}$ and $y=a^{\cos^{-1}t/2}$ ; differentiating and dividing gives $$-y/x$$ .

Answer:

$\dfrac{dy}{dx}=-\dfrac yx$ .

7Exercise 5.717 questions

Q.1Find the second order derivative of

$x^2+3x+2$

Solution

Differentiate twice.

Answer:

$$2$$ .

Q.2Find the second order derivative of

x^{20}

Solution

$y'=20x^{19}$ and $y''=380x^{18}$ .

Answer:

$380x^{18}$ .

Q.3Find the second order derivative of

x\cos x

Solution

Use the product rule twice.

Answer:

$-2\sin x-x\cos x$ .

Q.4Find the second order derivative of

\log x

Solution

$$y'=1/x$$ , so $$y''=-1/x^2$$ .

Answer:

$-\dfrac1{x^2}$ .

Q.5Find the second order derivative of

x^3\log x

Solution

$y'=3x^2\log x+x^2$ , then differentiate again.

Answer:

$6x\log x+5x$ .

Q.6Find the second order derivative of

e^x\sin5x

Solution

Differentiate by the product and chain rules twice.

Answer:

$e^x(10\cos5x-24\sin5x)$ .

Q.7Find the second order derivative of

e^{6x}\cos3x

Solution

Differentiate twice and collect sine and cosine terms.

Answer:

$e^{6x}(27\cos3x-36\sin3x)$ .

Q.8Find the second order derivative of

\tan^{-1}x

Solution

Differentiate $$1/(1+x^2)$$ .

Answer:

$-\dfrac{2x}{(1+x^2)^2}$ .

Q.9Find the second order derivative of

\log(\log x)

Solution

Differentiate $1/(x\log x)$ .

Answer:

$-\dfrac{\log x+1}{x^2(\log x)^2}$ .

Q.10Find the second order derivative of

\sin(\log x)

Solution

Differentiate $\cos(\log x)/x$ .

Answer:

$-\dfrac{\sin(\log x)+\cos(\log x)}{x^2}$ .

Q.11If

y=5\cos x-3\sin x

, prove that

\dfrac{d^2y}{dx^2}+y=0

Solution

$y''=-5\cos x+3\sin x=-y$ .

Answer:

$$y''+y=0$$ .

Q.12If

y=\cos^{-1}x

, find

\dfrac{d^2y}{dx^2}

in terms of

$y$

alone.v

Solution

$y''=-x/(1-x^2)^{3/2}$ ; substitute $x=\cos y$ .

Answer:

$-\dfrac{\cos y}{\sin^3y}$ .

Q.13If

y=3\cos(\log x)+4\sin(\log x)

, show that

$x^2y_2+xy_1+y=0$

Solution

Let $t=\log x$ . Then $$xy_1=dy/dt$$ and $$x^2y_2+xy_1=d^2y/dt^2=-y$$ .

Answer:

$$x^2y_2+xy_1+y=0$$ .

Q.14If

y=Ae^{mx}+Be^{nx}

, show that

\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0

Solution

Each term $e^{mx}$ and $e^{nx}$ separately satisfies the auxiliary expression, so their linear combination does too.

Answer:

$$y''-(m+n)y'+mny=0$$ .

Q.15If

y=500e^{7x}+600e^{-7x}

, show that

\dfrac{d^2y}{dx^2}=49y

Solution

Differentiating twice multiplies both exponential terms by $$49$$ .

Answer:

$$y''=49y$$ .

Q.16If

$e^y(x+1)=1$

, show that

\dfrac{d^2y}{dx^2}=\left(\dfrac{dy}{dx}\right)^2

Solution

From $$e^y(x+1)=1$$ , $y=-\log(x+1)$ , so $$y'=-1/(x+1)$$ and $$y''=1/(x+1)^2$$ .

Answer:

$$y''=(y')^2$$ .

Q.17If

y=(\tan^{-1}x)^2

, show that

$(x^2+1)^2y_2+2x(x^2+1)y_1=2$

Solution

$y_1=2\tan^{-1}x/(1+x^2)$ . Differentiating and substituting in the left side simplifies to $$2$$ .

Answer:

$$(x^2+1)^2y_2+2x(x^2+1)y_1=2$$ .

8Miscellaneous Exercise22 questions

Q.1Differentiate

$(3x^2-9x+5)^9$

w.r.t.

$x$

Solution

Use the chain rule with inner function $$3x^2-9x+5$$ .

Answer:

$$9(3x^2-9x+5)^8(6x-9)$$ .

Q.2Differentiate

\sin^3x+\cos^6x

w.r.t.

$x$

Solution

Differentiate each power by the chain rule.

Answer:

$3\sin^2x\cos x-6\cos^5x\sin x$ .

Q.3Differentiate

(5x)^{3\cos2x}

w.r.t.

$x$

Solution

Use logarithmic differentiation for $$u^v$$ with $$u=5x$$ , $v=3\cos2x$ .

Answer:

$(5x)^{3\cos2x}\left[-6\sin2x\log(5x)+\dfrac{3\cos2x}{x}\right]$ .

Q.4Differentiate

\sin^{-1}(x\sqrt{x})

0\le x\le1

, w.r.t.

$x$

Solution

Since $x\sqrt{x}=x^{3/2}$ , apply the chain rule to $\sin^{-1}(x^{3/2})$ .

Answer:

$\dfrac{3\sqrt{x}}{2\sqrt{1-x^3}}$ .

Q.5Differentiate

\dfrac{\cos^{-1}(x/2)}{\sqrt{2x+7}}

-2\lt x\lt2

, w.r.t.

$x$

Solution

Write the function as $\cos^{-1}(x/2)(2x+7)^{-1/2}$ and use the product rule.

Answer:

$-\dfrac{1}{2\sqrt{1-x^2/4}\sqrt{2x+7}}-\dfrac{\cos^{-1}(x/2)}{(2x+7)^{3/2}}$ .

Q.6Differentiate

\cot^{-1}\left[\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]

0\lt x\lt\dfrac\pi2

, w.r.t.

$x$

Solution

For $0\lt x\lt\pi/2$ , the expression inside $\cot^{-1}$ simplifies to $\cot(x/2)$ , so the function is $$x/2$$ .

Answer:

$\dfrac12$ .

Q.7Differentiate

(\log x)^{\log x}

x\gt1

, w.r.t.

$x$

Solution

Logarithmic differentiation gives $\log y=(\log x)\log(\log x)$ .

Answer:

$(\log x)^{\log x}\dfrac{\log(\log x)+1}{x}$ .

Q.8Differentiate

\cos(a\cos x+b\sin x)

, for constants

$a$

and

$b$

, w.r.t.

$x$

Solution

Apply the chain rule; the derivative of the inner function is $-a\sin x+b\cos x$ .

Answer:

$(a\sin x-b\cos x)\sin(a\cos x+b\sin x)$ .

Q.9Differentiate

(\sin x-\cos x)^{(\sin x-\cos x)}

\dfrac\pi4\lt x\lt\dfrac{3\pi}{4}

, w.r.t.

$x$

Solution

Let $u=\sin x-\cos x$ . Then $$y=u^u$$ and $y'=u^u u'(\log u+1)$ .

Answer:

$(\sin x-\cos x)^{(\sin x-\cos x)}(\sin x+\cos x)[\log(\sin x-\cos x)+1]$ .

Q.10Differentiate

$x^x+x^a+a^x+a^a$

, for fixed

a\gt0

and

x\gt0

, w.r.t.

$x$

Solution

Differentiate each term; $$a^a$$ is constant.

Answer:

$x^x(1+\log x)+ax^{a-1}+a^x\log a$ .

Q.11Differentiate

x^{x^2-3}+(x-3)^{x^2}

, for

x\gt3

, w.r.t.

$x$

Solution

Use logarithmic differentiation separately on the two variable-power terms.

Answer:

$x^{x^2-3}\left(2x\log x+\dfrac{x^2-3}{x}\right)+(x-3)^{x^2}\left(2x\log(x-3)+\dfrac{x^2}{x-3}\right)$ .

Q.12Find

\dfrac{dy}{dx}

y=12(1-\cos t)

x=10(t-\sin t)

-\dfrac\pi2\lt t\lt\dfrac\pi2

Solution

$dy/dt=12\sin t$ and $dx/dt=10(1-\cos t)$ ; divide and use $\sin t/(1-\cos t)=\cot(t/2)$ .

Answer:

$\dfrac65\cot\dfrac t2$ .

Q.13Find

\dfrac{dy}{dx}

y=\sin^{-1}x+\sin^{-1}\sqrt{1-x^2}

0\lt x\lt1

Solution

For $0\lt x\lt1$ , $\sin^{-1}\sqrt{1-x^2}=\dfrac\pi2-\sin^{-1}x$ , so $$y$$ is constant.

Answer:

$$0$$ .

Q.14If

x\sqrt{1+y}+y\sqrt{1+x}=0

-1\lt x\lt1

, prove that

\dfrac{dy}{dx}=-\dfrac1{(1+x)^2}

Solution

The equation is satisfied by $$y=-x/(1+x)$$ on the given interval. Differentiating gives $$y'= -[(1+x)-x]/(1+x)^2=-1/(1+x)^2$$ .

Answer:

$\dfrac{dy}{dx}=-\dfrac1{(1+x)^2}$ .

Q.15If

$(x-a)^2+(y-b)^2=c^2$

, for

c\gt0

, prove that

\dfrac{\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}}{\dfrac{d^2y}{dx^2}}

is a constant independent of

$a$

and

$b$

Solution

Implicit differentiation gives $$y'=-(x-a)/(y-b)$$ and $$y''=-c^2/(y-b)^3$$ . Also $$1+(y')^2=c^2/(y-b)^2$$ , so the given ratio is constant on each branch.

Answer:

The expression is constant on a fixed branch of the circle and is independent of $$a$$ and $$b$$ .

Q.16If

\cos y=x\cos(a+y)

, with

\cos a\ne\pm1

, prove that

\dfrac{dy}{dx}=\dfrac{\cos^2(a+y)}{\sin a}

Solution

Differentiating gives $y'[x\sin(a+y)-\sin y]=\cos(a+y)$ . Using $x=\cos y/\cos(a+y)$ , the bracket becomes $\sin a/\cos(a+y)$ .

Answer:

$\dfrac{dy}{dx}=\dfrac{\cos^2(a+y)}{\sin a}$ .

Q.17If

x=a(\cos t+t\sin t)

and

y=a(\sin t-t\cos t)

, find

\dfrac{d^2y}{dx^2}

Solution

First $dy/dx=\tan t$ . Then $d^2y/dx^2=(d(\tan t)/dt)/(dx/dt)=\sec^2t/(at\cos t)$ .

Answer:

$\dfrac1{at\cos^3t}$ .

Q.18If

$f(x)=|x|^3$

, show that

$f''(x)$

exists for all real

$x$

and find it.v

Solution

For $x\gt0$ , $$f=x^3$$ ; for $x\lt0$ , $$f=-x^3$$ ; at $$0$$ , $$f'(x)=3x|x|$$ gives $$f''(0)=0$$ .

Answer:

$$f''(x)=6|x|$$ for all real $$x$$ .

Q.19Using

\sin(A+B)=\sin A\cos B+\cos A\sin B

and differentiation, obtain the sum formula for cosines.v

Solution

Differentiate the sine addition formula with respect to $$A$$ , treating $$B$$ as constant.

Answer:

$\cos(A+B)=\cos A\cos B-\sin A\sin B$ .

Q.20Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.v

Solution

The sum of two absolute-value functions is continuous everywhere and fails to be differentiable exactly at its two vertices $$a$$ and $$b$$ .

Answer:

Yes. For example, $$f(x)=|x-a|+|x-b|$$ with $a\ne b$ .

Q.21If

y=\begin{vmatrix}f(x)&g(x)&h(x)\\l&m&n\\a&b&c\end{vmatrix}

, prove that

\dfrac{dy}{dx}=\begin{vmatrix}f'(x)&g'(x)&h'(x)\\l&m&n\\a&b&c\end{vmatrix}

Solution

Expand the determinant along the first row. The cofactors are constants, so differentiating affects only $$f(x),g(x),h(x)$$ .

Answer:

$\dfrac{dy}{dx}=\begin{vmatrix}f'(x)&g'(x)&h'(x)\\l&m&n\\a&b&c\end{vmatrix}$ .

Q.22If

y=e^{a\cos^{-1}x}

-1\le x\le1

, show that

(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}-a^2y=0

Solution

Since $y'=-ay/\sqrt{1-x^2}$ , differentiating once more and substituting in $$(1-x^2)y''-xy'-a^2y$$ makes all terms cancel.

Answer:

$$(1-x^2)y''-xy'-a^2y=0$$ .

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