• Home
  • Textbooks
  • A Textbook of Differential Calculus
  • Differentiation

A Textbook of Differential Calculus

Amit M Agarwal

Chapter 2

Differentiation - all with Video Answers

Educators

Chapter Questions

01:53

Problem 1

If $y=a t^{2}+2 b t+c$ and $t=a x^{2}+2 b x+c$, then $\frac{d^{3} y}{d x^{3}}$ equals to
(a) $24 a^{2}(a t+b)$
(b) $24 a(a x+b)^{2}$
(c) $24 a(a t+b)^{2}$
(d) $24 a^{2}(a x+b)$

Kamalesh Kumar
Kamalesh Kumar
Numerade Educator
02:06

Problem 2

Differential coefficient of $\left(x^{\frac{1+m}{m-n}}\right)^{\frac{1}{n-l}} \cdot\left(x^{\frac{m+n}{n-l}}\right)^{\frac{1}{l-m}} \cdot\left(x^{\frac{n+l}{l-m}}\right)^{\frac{1}{m-n}}$ w.r.t. $x$, is
(a) 1
(b) 0
(c) $-1$
(d) $x^{l m n}$
ots .$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:59

Problem 3

If $y=(A+B x) e^{m x}+(m-1)^{-2} e^{x}$, then $\frac{d^{2} y}{d x^{2}}-2 m \frac{d y}{d x}+m^{2} y$ is equal to
(a) $e^{x}$
(b) $e^{m x}$
(c) $e^{-m x}$
(d) $e^{(1-m) x}$

Urvashi Arora
Urvashi Arora
Numerade Educator
01:40

Problem 4

Given, $f(x)=-\frac{x^{3}}{3}+x^{2} \sin 1.5 a-x \sin a \cdot \sin 2 a-5 \arcsin \left(a^{2}-8 a+17\right)$, then
(a) $f(x)$ is not defined at $x=\sin 8$
(b) $f^{\prime}(\sin 8)>0$
(c) $f^{\prime}(x)$ is not defined at $x=\sin 8$
(d) $f^{\prime}(\sin 8)<0$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:45

Problem 5

If $f$ and $g$ are differentiable functions such that $g^{\prime}(a)=2$ and $g(a)=b$ and if $f o g$ is an identity function, then $f^{\prime}(b)$ has the value equal to
(a) $2 / 3$
(b) 1
(c) 0
(d) $1 / 2$

Aman Gupta
Aman Gupta
Numerade Educator
01:12

Problem 6

The derivative of the function, $f(x)=\cos ^{-1}\left\{\frac{1}{\sqrt{13}}(2 \cos x-3 \sin x)\right\}+\sin ^{-1}\left\{\frac{1}{\sqrt{13}}(2 \cos x+3 \sin x)\right\}$
w.r.t. $\sqrt{1+x^{2}}$ at $x=\frac{3}{4}$ is
(a) $\frac{3}{2}$
(b) $\frac{5}{2}$
(c) $\frac{10}{3}$
(d) 0

Gio Maya
Gio Maya
Numerade Educator
02:48

Problem 7

If $f(x)=\sqrt{x+2 \sqrt{2 x-4}}+\sqrt{x-2 \sqrt{2 x-4}}$, then the value of $10 f^{\prime}\left(102^{+}\right)$, is
(a) $-1$
(b) 0
(c) $\underline{1}$
(d) does not exist

Srilakshmi E K
Srilakshmi E K
Numerade Educator
01:45

Problem 8

Let $y=\ln (1+\cos x)^{2}$, then the value of $\frac{d^{2} y}{d x^{2}}+\frac{2}{e^{y / 2}}$ equals to
(a) 0
(b) $\frac{2}{1+\cos x}$
(c) $\frac{4}{(1+\cos x)}$
(d) $\frac{-4}{(1+\cos x)^{2}}$

Adriano Chikande
Adriano Chikande
Numerade Educator
03:16

Problem 9

If $f(x)=\frac{a+\sqrt{a^{2}-x^{2}}+x}{\sqrt{a^{2}-x^{2}}+a-x}$ where $a>0$ and $x<a$, then $f^{\prime}(0)$ has the value equal to
(a) $\sqrt{a}$
(b) $a$
(c) $\frac{1}{\sqrt{a}}$
(d) $\frac{1}{a}$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:43

Problem 10

Let $u(x)$ and $v(x)$ are differentiable functions such that $\frac{u(x)}{v(x)}=7$. If $\frac{u^{\prime}(x)}{d^{\prime}(x)}=p$ and $\left(\frac{u(x)}{v(x)}\right)=q$, then $\frac{p+q}{p-q}$ has the value equal to
(a) 1
(b) 0
(c) 7
(d) $-7$

Lucas Finney
Lucas Finney
Numerade Educator
04:19

Problem 11

If $f(x)=\left|\log _{e}\right| x \|$, then $f^{\prime}(x)$ equals to
(a) $\frac{1}{|x|}, x \neq 0$
(b) $\frac{1}{x}$ for $|x|>1$ and $-\frac{1}{x}$ for $|x|<1$
(c) $-\frac{1}{x}$ for $|x|>1$ and $\frac{1}{x}$ for $|x|<1$
(d) $\frac{1}{x}$ for $x>0$ and $-\frac{1}{x}$ for $x<0$

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
02:15

Problem 12

If $f(x)$ is given by $f(x)=(\cos x+i \sin x)(\cos 3 x+i \sin 3 x) \ldots$
$\ldots(\cos (2 n-1) x+i \sin (2 n-1) x)$
then $f^{\prime \prime}(x)$ is equal to
(a) $n^{3} f(x)$
(b) $-n^{4} f(x)$
(c) $-n^{2} f(x)$
(d) $n^{4} f(x)$

Amy Jiang
Amy Jiang
Numerade Educator
01:31

Problem 13

Let $f(x)=x^{n}, n$ being a non-negative integer. The value of $n$ for which the equality $f^{\prime}(x+y)=f^{\prime}(x)+f^{\prime}(y)$ is valid for all $x, y>0$, is
(a) 0,1
(b) 1,2
(c) 2,4
(d) None of these

Aman Gupta
Aman Gupta
Numerade Educator
05:40

Problem 14

If $y=\sin ^{-1}\left(\frac{\sin \alpha \sin x}{1-\cos \alpha \sin x}\right)$, then $y^{\prime}(0)$ is
(a) 1
(b) $2 \tan \alpha$
(c) $\frac{1}{2} \tan \alpha$
(d) $\sin \alpha$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:47

Problem 15

If $f(x)=\sin \left\{\frac{\pi}{3}[x]-x^{2}\right\}$ for $2<x<3$ and $[x]$ denotes the greatest integer less than or equal to $x$, then $f^{\prime}(\sqrt{\pi / 3})$ is equal to
(a) $\sqrt{\pi / 3}$
(b) $-\sqrt{\pi / 3}$
(c) $-\sqrt{\pi}$
(d) None of these

Ankit Singh
Ankit Singh
Numerade Educator
01:06

Problem 16

The functions $u=e^{x} \sin x, v=e^{x} \cos x$ satisfy the equation
(a) $v \frac{d u}{d x}-u \frac{d v}{d x}=u^{2}+v^{2}$
(b) $\frac{d^{2} u}{d x^{2}}=2 v$
(c) $\frac{d^{2} v}{d x^{2}}=-2 u$
(d) All of these

Carson Merrill
Carson Merrill
Numerade Educator
01:26

Problem 17

If $f(x)=\log _{x}\{\ln (x)\}$, then $f^{\prime}(x)$ at $x=e$, is
(a) $e$
(b) $-e$
(c) $e^{2}$
(d) $e^{-1}$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
01:10

Problem 18

Let $f$ be a differentiable function satisfying $[f(x)]^{n}=f(n x)$ for all $x \in R$. Then, $f^{\prime}(x) f(n x)$
(a) $f(x)$
(b) 0
(c) $f(x) f^{\prime}(n x)$
(d) None of these

Aman Gupta
Aman Gupta
Numerade Educator
06:00

Problem 19

If $y=f(x)$ is an odd differentiable function defined on $(-\infty, \infty)$ such that $f^{\prime}(3)=-2$, then $f^{\prime}(-3)$ equals
(a) 4
(b) 2 $\begin{array}{ll}\text { (c) }-2 & \text { (d) } 0\end{array}$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:25

Problem 20

If $y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\ldots \infty}}}}$, then $\frac{d y}{d x}$ is equal to
(a) $\frac{y+x}{y^{2}-2 x}$
(b) $\frac{y^{3}-x}{2 y^{2}-2 x y-1}$
(c) $\frac{y^{3}+x}{2 y^{2}-x}$
(d) None of these

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
05:39

Problem 21

If $y=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \ldots\left(1+x^{2^{n}}\right)$, then $\frac{d y}{d x}$ at $x=0$ is
(a) 1
(b) $-1$
(c) 0
(d) None of these

SA
Safina Aman
Numerade Educator
02:52

Problem 22

If $f(x)=|\cos x-\sin x|$ then $f^{\prime}\left(\frac{\pi}{4}\right)$ is equal to
(a) $\sqrt{2}$
(b) $-\sqrt{2}$
(c) 0
(d) None of these

Aman Gupta
Aman Gupta
Numerade Educator
01:32

Problem 23

Let $f(x)=x^{2}+x g^{\prime}(1)+g^{\prime \prime}(2)$ and $g(x)=x^{2}+x f^{\prime}(2)+f^{\prime \prime}(3)$. Then,
(a) $f^{\prime}(1)=4+f^{\prime}(2)$
(b) $g^{\prime}(2)=8+g^{\prime}(1)$
(c) $g^{\prime \prime}(2)+f^{\prime \prime}(3)=4$
(d) All of these

Allison Knapp
Allison Knapp
Numerade Educator
01:14

Problem 24

If $f(x)=x^{n}$, then the value of $f(1)-\frac{f^{\prime}(1)}{1 !}+\frac{f^{\prime \prime}(1)}{2 !}-\frac{f^{\prime \prime \prime}(1)}{3 !}+\frac{f^{(\omega)}(1)}{4 !}-\ldots$
$\ldots+\frac{(-1)^{n} f^{n}(1)}{n !}$ is
(a) 1
(b) $2^{n}$
(c) $2^{n-1}$
(d) 0

Aman Gupta
Aman Gupta
Numerade Educator
01:55

Problem 25

If $y=x^{(\ln x)^{\operatorname{lm} \ln x \mid}}$, then $\frac{d y}{d x}$ is equal to
(a) $\frac{y}{x}\left(\ln x^{\ln x-1}+2 \ln x \ln (\ln x)\right)$
(b) $\frac{y}{x}(\ln x)^{\ln (\ln x)}(2 \ln (\ln x)+1)$
(c) $\frac{y}{x \ln x}\left((\ln x)^{2}+2 \ln (\ln x)\right)$
(d) $\frac{y}{x} \cdot \frac{\ln y}{\ln x}(2 \ln (\ln x)+1)$

Kamalesh Kumar
Kamalesh Kumar
Numerade Educator
06:20

Problem 26

Which of the following functions are not derivable at $x=0$ ?
(a) $f(x)=\sin ^{-1} 2 x \sqrt{1-x^{2}}$
(b) $g(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^{x}}\right)$
(c) $h(x)=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$
(d) $k(x)=\sin ^{-1}(\cos x)$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:21

Problem 27

Let $f(x)=\frac{\sqrt{x-2 \sqrt{x-1}}}{\sqrt{x-1}-1} \cdot x$, then
(a) $f^{\prime}(10)=1$
(b) $f^{\prime}(3 / 2)=-1$
(c) domain of $f(x)$ is $x \geq 1$
(d) None of these

Pawan Yadav
Pawan Yadav
Numerade Educator
01:28

Problem 28

if $2^{x}+2^{y}=2^{x+y}$, then $\frac{d y}{d x}$ has the value equal to
(a) $-\frac{2^{y}}{2^{x}}$
(b) $\frac{1}{1-2^{x}}$
(c) $1-2^{y}$
(d) $\frac{2^{x}\left(1-2^{y}\right)}{2^{y}\left(2^{x}-1\right)}$

Aman Gupta
Aman Gupta
Numerade Educator
08:37

Problem 29

for the function $y=f(x)=\left(x^{2}+b x+c\right) e^{x}$, which of the following holds?
(a) If $f(x)>0$ for all real $x \Rightarrow f^{\prime}(x)>0$
(b) If $f(x)>0$ for all real $x \Rightarrow f^{\prime}(x)>0$
(c) If $f^{\prime}(x)>0$ for all real $x \Rightarrow f(x)>0$
(d) If $f^{\prime}(x)>0$ for all real $x \neq f(x)>0$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:25

Problem 30

If $\sqrt{y+x}+\sqrt{y-x}=c$ (where $c \neq 0$ ), then $\frac{d y}{d x}$ has the value equal to
(a) $\frac{2 x}{c^{2}}$
(b) $\frac{x}{y+\sqrt{y^{2}-x^{2}}}$
(c) $\frac{y-\sqrt{y^{2}-x^{2}}}{x}$
(d) $\frac{c^{2}}{2 y}$

Aman Gupta
Aman Gupta
Numerade Educator
01:03

Problem 31

If $y=\tan x \tan 2 x \tan 3 x$, then $\frac{d y}{d x}$ has the value equal to
(a) $3 \sec ^{2} 3 x \tan x \tan 2 x+\sec ^{2} x \tan 2 x \tan 3 x+2 \sec ^{2} 2 x \tan 3 x \tan x$
(b) $2 y(\operatorname{cosec} 2 x+2 \operatorname{cosec} 4 x+3 \operatorname{cosec} 6 x)$
(c) $3 \sec ^{2} 3 x-2 \sec ^{2} 2 x-\sec ^{2} x$
(d) $\sec ^{2} x+2 \sec ^{2} 2 x+3 \sec ^{2} 3 x$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:53

Problem 32

The relation between the parameter ' $t$ ' and the angle $\alpha$ between the tangent to the given curve and the $x$-axis is given by, ' $t$ ' equals to
(a) $\frac{\pi}{2}-\alpha$
(b) $\frac{\pi}{4}+\alpha$
(c) $\alpha-\frac{\pi}{4}$
(d) $\frac{\pi}{4}-\alpha$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:54

Problem 33

The value of $\frac{d^{2} y}{d r^{2}}$ at the point where $t=0$ is
(a) 1
(b) $\underline{2}$
(c) $-2$
(d) 3

Gaurav Kumar
Gaurav Kumar
Numerade Educator
02:36

Problem 34

If $F(t)=\int(x+y) d t$, then the value of $F\left(\frac{\pi}{2}\right)-F(0)$ j
(a) 1
(b) $-1$
(c) $e^{\pi / 2}$
(d) 0

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:33

Problem 35

If $x-c$ is a factor of order $m$ of the polynomial $f(x)$ of degree $n(1<m<n)$, then $x=c$ is a root of the polynomial (where $f^{r}(x)$ represents $r$ th derivative of $f(x)$
W.r.t. $x)$
(a) $f^{m}(x)$
(b) $f^{m-1}(x)$
(c) $f^{n}(x)$
(d) None of these

Aman Gupta
Aman Gupta
Numerade Educator
05:51

Problem 36

If $a_{1} x^{3}+b_{1} x^{2}+c_{1} x+d_{1}=0$ and $a_{2} x^{3}+b_{2} x^{2}+c_{2} x+d_{2}=0$ have a pair of repeated common roots, then
$$
\left|\begin{array}{ccc}
3 a_{1} & 2 b_{1} & c_{1} \\
3 a_{2} & 2 b_{2} & c_{2} \\
a_{2} b_{1}-a_{1} b_{2} & c_{1} a_{2}-c_{2} a_{1} & d_{1} a_{2}-d_{2} a_{1}
\end{array}\right| \text { is }
$$
(a) 0
(b) $c_{1} a_{2}-c_{2} a_{1}$
(c) $a_{1} b_{2}-a_{2} b_{1}$
(d) $d_{1} a_{2}-d_{2} a_{1}$

Anas Venkitta
Anas Venkitta
Numerade Educator
00:58

Problem 37

If $\alpha$ occurs $p$ times and $\beta$ occurs $q$ times in polynomial equation $f(x)=0$ of degree $n(1<p, q<n)$, then which of the following is not true? (where $f^{r}(x)$ represents $r$ th derivative of $f(x)$ w.r.t. $x$ )
(a) If $p<q<n$, then $\alpha$ and $\beta$ are two of the roots of the equation $f^{p-1}(x)=0$.
(b) If $q<p<n$, then $\alpha$ and $\beta$ are two of the roots of the equation $f^{q-1}(x)=0$.
(c) If $p<q<n$, then equations $f(x)=0$ and $f^{q-1}(x)=0$ have exactly one root common.
(d) If $q<p<n$, then equations $f^{q}(x)=0$ and $f^{p}(x)=0$ have exactly two roots common.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:40

Problem 38

The value of $\left(1-a_{1}\right)\left(1-a_{2}\right) \ldots\left(1-a_{n}\right)$, is
(a) $\frac{n^{2}}{2}$
(b) $\underline{n}$
(c) $(-1)^{n} n$
(d) None of these

Srilakshmi E K
Srilakshmi E K
Numerade Educator
03:02

Problem 39

The value of $\sum_{r=2}^{n} \frac{1}{2-a_{r}}$, is
(a) $\frac{2^{n-1}(n-2)+1}{2^{n}-1}$
(b) $\frac{2^{n}(n-2)+1}{2^{n}-1}$
(c) $\frac{2^{n-1}(n-1)-1}{2^{n}-1}$
(d) None of these

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:02

Problem 40

The value of $\sum_{r=2}^{n} \frac{1}{1-a_{r}}$, is
(a) $\frac{n}{4}$
(b) $\frac{n(n-1)}{2}$
(c) $\frac{n-1}{2}$
(d) None of these

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:17

Problem 41

Match the entries between following two columns:
Column I
(A) $y=f(x)$ be given by $x=t^{5}-5 t^{3}-20 t+7, \quad$ Column II
$y=4 t^{3}-3 t^{2}-18 t+3$, then $-5 \times \frac{d y}{d x}$ at $t=1$
(B) $P(x)$ be a polynomial of degree 4 with $P(2)=-1, P^{\prime}(2)=0$,
$P^{\prime \prime}(2)=2, P^{\prime \prime}(2)=-12$ and $P^{i v}(2)=24$, then $P^{\prime \prime}(3)$ is equal to
(C) $y=\frac{1}{x}$, then $\frac{\frac{d y}{\sqrt{1+y^{4}}}}{\frac{d x}{\sqrt{1+x^{4}}}}$
(D) $f\left(\frac{2 x+3 y}{5}\right)=\frac{2 f(x)+3 f(y)}{5} \operatorname{and} f^{\prime}(0)=p$ and $f(0)=q$, then $f^{\prime \prime}(0)$
(s) $-1$

Lucas Finney
Lucas Finney
Numerade Educator
04:56

Problem 42

Match the following :
\begin{tabular}{l|l}
\hline Column I & Column II \\
\hline (A) $y=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$, then $\frac{d y}{d x}=-\frac{2}{1+x^{2}}$ & (p) for $x<0$ \\
(B) $y=\cos ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)$, then $\frac{d y}{d x}=\frac{1}{1+x^{2}}$ & (q) for $x>1$ \\
(C) $y=\left|e^{|x|}-e\right|$, then $\frac{d y}{d x}>0$ & (r) for $x<-1$ \\
(D) $u=\log |2 x|, v=\left|\tan ^{-1} x\right|$, then $\frac{d u}{d v}>2$ & (s) for $-1<x<0$ \\
\hline
\end{tabular}

Gaurav Kumar
Gaurav Kumar
Numerade Educator
01:58

Problem 43

Suppose, $A=\frac{d y}{d x}$ of $x^{2}+y^{2}=4$ at $(\sqrt{2}, \sqrt{2}), B=\frac{d y}{d x}$ of $\sin y+\sin x=\sin x \cdot \sin y$ at
$(\pi, \pi)$ and $C=\frac{d y}{d x}$ of $2 e^{x y}+e^{x} e^{y}-e^{x}-e^{y}=e^{x y+1}$ at $(1,1)$, then $(A-B-C)$ has
the value equal to .........

Malika Singh
Malika Singh
Numerade Educator
00:35

Problem 44

A function is represented parametrically by the equations $x=\frac{1+t}{t^{3}}$; $y=\frac{3}{2 t^{2}}+\frac{2}{t}$, then $\frac{d y}{d x}-x \cdot\left(\frac{d y}{d x}\right)^{3}$ has the absolute value equal to $\ldots \ldots \ldots .$

Tony Ni
Tony Ni
Numerade Educator
00:34

Problem 45

Suppose, the function $f(x)-f(2 x)$ has the derivative 5 at $x=1$ and derivative 7 at $x=2$. The derivative of the function $f(x)-f(4 x)$ at $x=1$, has the value $10+\lambda$, then the value of $\lambda$ is equal to $\ldots \ldots \ldots$

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
07:35

Problem 46

If $x+y=3 e^{2}$, then $D\left(x^{y}\right)$ vanishes when $x$ equals to $\lambda e^{2}$, then the value of $\lambda$ is equal to .........

Jack Chen
Jack Chen
Numerade Educator
01:29

Problem 47

Let $h(x)$ be differentiable for all $x$ and let $f(x)=\left(k x+e^{x}\right) h(x)$ where $k$ is some constant. If $h(0)=5, h^{\prime}(0)=-2$ and $f^{\prime}(0)=18$, then the value of $k$ is equal to $\ldots \ld

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator