Question 1: Palette 1/20

DPT MCQ

If $f(x)=x^n$, where $n\neq1$, then using the first principle, $f'(x)$ is:

A

$nx^{n-1}$

B

$nx^n$

C

$n x^{n+1}$

D

$x^{n-1}$

Question 2: Palette 2/20

DPT MCQ

If $f(x)=a^x$, where $a>0$ and $a\neq1$, then using the first principle, $f'(x)$ is:

A

$a^x$

B

$a^x\ln a$

C

$x a^{x-1}$

D

$\ln a$

Question 3: Palette 3/20

DPT MCQ

If $f(x)=\sin x$, then using the first principle, $f'(x)$ is:

A

$\sin x$

B

$-\cos x$

C

$\cos x$

D

$-\sin x$

Question 4: Palette 4/20

DPT MCQ

If $D(y)$ denotes derivative with respect to $x$, find $D(e^\pi)$ and $D(x^e)$.

A

$0, e x^{e-1}$

B

$e^\pi, e x^{e-1}$

C

$0, x^{e-1}$

D

$e^\pi, e x^e$

Question 5: Palette 5/20

DPT MCQ

Find $D\left(\frac{1}{\sin x}\right)$.

A

$-\csc x\cot x$

B

$\csc x\cot x$

C

$-\sec x\tan x$

D

$\csc^2x$

Question 6: Palette 6/20

DPT MCQ

Find $D\left(\frac{\sin 2x}{1+\cos 2x}\right)$.

A

$\sec^2x$

B

$-\csc^2x$

C

$\tan x$

D

$\cos 2x$

Question 7: Palette 7/20

DPT MCQ

Find $D(\tan(\tan^{-1}x))$.

A

$1$

B

$0$

C

$\frac{1}{1+x^2}$

D

$\sec^2x$

Question 8: Palette 8/20

DPT MCQ

If $y=\left(\frac{x^a}{x^b}\right)^{a+b}\left(\frac{x^b}{x^c}\right)^{b+c}\left(\frac{x^c}{x^a}\right)^{c+a}$, find $\frac{dy}{dx}$.

A

$0$

B

$1$

C

$x$

D

$a+b+c$

Question 9: Palette 9/20

DPT MCQ

If $y=e^x+3\ln x-4\tan^{-1}x+\sin 3x+4\sin^3x$, find $\frac{dy}{dx}$.

A

$e^x+\frac{3}{x}-\frac{4}{1+x^2}+3\cos x$

B

$e^x+\frac{3}{x}-\frac{4}{1+x^2}+3\cos x+12\sin^2x$

C

$e^x+\frac{3}{x}+\frac{4}{1+x^2}+3\cos x$

D

$e^x+\frac{3}{x}-\frac{4}{1+x^2}-3\cos x$

Question 10: Palette 10/20

DPT MCQ

Find the derivative with respect to $x$ of $y=e^x+3\ln x-4\sin x$.

A

$e^x+\frac{3}{x}-4\cos x$

B

$e^x+\frac{3}{x}+4\cos x$

C

$e^x+\frac{3}{x}-4\sin x$

D

$e^x+\frac{3}{x^2}-4\cos x$

Question 11: Palette 11/20

DPT MCQ

Find the derivative with respect to $x$ of $y=x\sin^{-1}x$.

A

$\sin^{-1}x+\frac{x}{\sqrt{1-x^2}}$

B

$\sin^{-1}x-\frac{x}{\sqrt{1-x^2}}$

C

$\frac{1}{\sqrt{1-x^2}}$

D

$x\sin^{-1}x+\frac{1}{\sqrt{1-x^2}}$

Question 12: Palette 12/20

DPT MCQ

Find the derivative with respect to $x$ of $y=x^2e^x\ln x$.

A

$x e^x\left((x+2)\ln x+1\right)$

B

$x e^x\left((x+2)\ln x-1\right)$

C

$2x e^x\ln x+x e^x$

D

$x^2e^x\ln x+x e^x$

Question 13: Palette 13/20

DPT MCQ

If $f(x)=(1+x)(3+x^2)^{1/2}(9+x^3)^{1/3}$, then $f'(-1)$ is equal to:

A

$0$

B

$2\sqrt{2}$

C

$4$

D

$6$

Question 14: Palette 14/20

DPT MCQ

If $y=\frac{x^3+x^2+1}{x^2+1}$, then $\frac{dy}{dx}$ is

A

$\frac{x^2(x^2+3)}{(x^2+1)^2}$

B

$\frac{x^4+3x}{(x^2+1)^2}$

C

$\frac{x^2(x^2+1)}{(x^2+1)^2}$

D

$\frac{3x^2+2x}{2x}$

Question 15: Palette 15/20

DPT MCQ

If $y=\frac{\sin^{-1}x-\cos^{-1}x}{\sin^{-1}x+\cos^{-1}x}$, then $\frac{dy}{dx}$ is

A

$\frac{4}{\pi\sqrt{1-x^2}}$

B

$\frac{2}{\pi\sqrt{1-x^2}}$

C

$\frac{1}{\pi\sqrt{1-x^2}}$

D

$\frac{-4}{\pi\sqrt{1-x^2}}$

Question 16: Palette 16/20

DPT MCQ

Let $f(x)=x^{1/3}\sin x$ and $f(0)=0$. Then $f'(0)$ is

A

$0$

B

$1$

C

Does not exist

D

Infinite

Question 17: Palette 17/20

DPT MCQ

$\frac{d}{dx}\tan(\log x)$ is

A

$\frac{\sec^2(\log x)}{x}$

B

$\sec^2x$

C

$\frac{\sec x}{x}$

D

$\tan(\log x)$

Question 18: Palette 18/20

DPT MCQ

$\frac{d}{dx}\sin(x^2)$ is

A

$\cos(x^2)$

B

$2x\cos(x^2)$

C

$2x\sin(x^2)$

D

$-2x\cos(x^2)$

Question 19: Palette 19/20

DPT MCQ

$\frac{d}{dx}\log(\sin(x^2))$ is

A

$\frac{2x}{\sin(x^2)}$

B

$2x\cot(x^2)$

C

$\cot(x^2)$

D

$2x\tan(x^2)$

Question 20: Palette 20/20

DPT MCQ

$\frac{d}{dx}\tan^{-1}(\log(\sin(x^2)))$ is

A

$\frac{2x\cot(x^2)}{1+(\log(\sin(x^2)))^2}$

B

$\frac{2x}{1+(\log(\sin(x^2)))^2}$

C

$\frac{\cot(x^2)}{1+(\log(\sin(x^2)))^2}$

D

$\frac{2x\sin(x^2)}{1+(\log(\sin(x^2)))^2}$