## Derivatives : part 2: IITJEE Maths : Tutorial problems for practice

Problem 1:

If $f(a)=2$, $f^{'}(a)=1$, $g(a)=-1$, $g^{'}(a)=2$, then the value of $\lim_{x \rightarrow a}\frac{g(x)f(a)-g(a)f(x)}{x-a}$ is

(a) -5 (b) $\frac{1}{5}$ (c) 5 (d) 0

Problem 2:

Let $y = \arcsin{(\frac{2x}{1+x^{2}})}$, $0 < x <1$ and $0 < y < \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to :

(a) $\frac{2}{1+x^{2}}$ (b) $\frac{2x}{1+x^{2}}$ (c) $\frac{-2}{1+x^{2}}$ (d) none

Problem 3:

Let $f(x) = ax^{2}+1$ for $x \leq 1$

and $f(x)= x+a$ for $x \leq 1$ then f is derivable at $x=1$, if

(a) $a=0$ (b) $a = \frac{1}{2}$ (c) $a=1$ (d) $a=2$

Problem 4:

If $f(x) = ax^{2}+b$ for $x \leq 1$

if $f(x)=b x^{2}+ax+c$ for $x>1$, where $b \neq 0$, then $f(x)$ is continuous and differentiable at $x=1$, if

(a) $c=0, a=2b$ (b) $a=2b, c \in \Re$ (c) $a=b, c=0$ (d) $a=2b, c \neq 0$

Problem 5:

$\lim_{h \rightarrow 0} \frac{\cos^{2}(x+h)- \cos^{2}(x)}{h}$ is equal to

(a) $\cos^{2}(x)$ (b) $-\sin{2x}$ (c) $\sin{x} \cos{x}$ (d) $2\sin{x}$

Problem 6:

$\lim_{h \rightarrow 0} \frac{\sin{\sqrt{x+h}-\sin{\sqrt{x}}}}{h}$ is equal to

(a) $\cos {\sqrt{x}}$ (b) $\frac{1}{2\sin{\sqrt{x}}}$ (c) $\frac{\cos{\sqrt{x}}}{2\sqrt{x}}$ (d) $\sin{\sqrt{x}}$

Problem 7:

$(\arccos{x})^{'}= \frac{-1}{\sqrt{1-x^{2}}}$ where

(a) $-1 < x <1$ (b) $-1 \leq x \leq 1$ (c) $-1 \leq x < 1$ (d) $-1 < x \leq 1$

Problem 8:

$\frac{d}{dx}(\arctan{(\frac{3x-x^{2}}{1-3x^{2}})})$ is equal to

(a) $\frac{3}{1+x^{2}}$ (b) $\frac{3}{1+9x^{2}}$ (c) $\sec^{2}{x}$ (d) $\frac{1}{9+x^{2}}$

Problem 9:

If $x=a\cos^{3}(t)$ and $y=a\sin^{3}(t)$, then $\frac{dy}{dx}$ is equal to

(a) $\cos{t}$ (b) $\cot{t}$ (c) $cosec{(t)}$ (d) $-\tan{t}$

Problem 10:

If $y = arcsin{\cos{x}}$, then $\frac{dy}{dx}$ is equal to

(a) -1 (b) $\cos{t}$ (c) $cosec{(t)}$ (d) $-\tan{t}$

Regards,

Nalin Pithwa

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