## Derivatives: part 7: IITJEE tutorial problems practice

Problem 1: Differential coefficient of $\sec{\arctan{x}}$ is

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

Problem 2: If $\sin{(x+y)} = \log{(x+y)}$, then $\frac{dy}{dx}$ is equal to :

(a) 2 (b) -2 (c) 1 (d) -1

Problem 3: If $y = \arcsin{\sqrt{x-ax}-\sqrt{a-ax}}$, then $\frac{dy}{dx}$ is equal to:

(a) $\frac{1}{2\sqrt{x}\sqrt{1-x}}$ (b) $\sin{(\sqrt{x})} \times \sin{(\sqrt{a})}$

(c) $\frac{1}{\sin{\sqrt{a-ax}}}$ (d) zero

Problem 4: For the differentiable function f, the value of : $\lim_{h \rightarrow 0} \frac{(f(x+h))^{2}-(f(x))^{2}}{2h}$ is equal to:

(a) $(f^{'}(x))^{2}$ (b) $\frac{1}{2}(f(x))^{2}$ (c) $f(x)f^{'}(x)$ (d) zero

Problem 5: The derivative of $\arctan{\frac{\sqrt{1+x^{2}}-1}{x}}$ w.r.t. $\arctan{(\frac{2x\sqrt{1-x^{2}}}{1-2x^{2}})}$ at $x=0$ is :

(a) $\frac{1}{8}$ (b) $\frac{1}{4}$ (c) $\frac{1}{2}$ (d) 1

Problem 6: If $x = e^{y+e^{y+e^{y+e^{y+ \ldots}}}}$ then $\frac{dy}{dx}$ is

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

Problem 7: Consider the following statements:

(1) $(\frac{f}{g})^{'} = \frac{f^{'}}{g^{'}}$ (2) $\frac{(fg)^{'}}{fg} = \frac{f^{'}}{f} + \frac{g^{'}}{g}$

(3) $\frac{(f+g)^{'}}{f+g} = \frac{f^{'}}{f} + \frac{g^{'}}{g}$ (4) $\frac{(f/g)^{'}}{f/g} = \frac{f^{'}}{f} + \frac{g^{'}}{g}$

Which of the following statements are true?

(a) 1 and 2 (b) 2 and 3 (c) 2 and 4 (d) 3 and 4

Problem 8: If $y=e^{x+3\log{x}}$ then $\frac{dy}{dx} =$

(a) $e^{x+3\log{x}}$ (b) $e^{x}.x^{2}(x+3)$ (c) $e^{x}. e^{3\log{x}}$ (d) $3x^{2}e^{x}$

Problem 9: If $y=\sin^{2}(x \deg)$, then find the value of $\frac{dy}{dx}$ is:

(a) $\frac{\pi}{360}\sin{(2 x \deg)}$ (b) $\frac{\pi}{2}\sin{(2x\deg)}$ (c) $180 \sin {(2x\deg)}$ (d) $\frac{\pi}{180}\sin{(2x\deg)}$

Problem 10: If $y=\log_{a}{x} + \log_{x}{a} + \log_{x}{x}+ \log_{a}{a}$ then the value of $\frac{dy}{dx}$ is:

(a) $\frac{1}{x}+x\log{a}$ (b) $\frac{\log{a}}{x} + \frac{x}{\log{a}}$ (c) $\frac{1}{x \log{a}}+ x \log{a}$ (d) $\frac{1}{x\log{x}} - \frac{\log{a}}{x(\log{x})^{2}}$

Cheers,

Nalin Pithwa.

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