Derivatives: part 8: IITJEE mains tutorial problems practice

Problem 1: If $y=b(\arctan{(\frac{x}{y})})+ \arctan{(\frac{y}{x})}$, then $\frac{dy}{dx}$ is equal to:

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

Problem 2: If $r=a(1+\cos{\theta})$, then $\sqrt{r^{2}+(\frac{dr}{d\theta})^{2}}$ is:

(a) $2a\cos{\theta}$ (b) $2a \sin{(\frac{\theta}{2})}$ (c) $2a \cos{(\frac{\theta}{2})}$ (d) $2a \sin{\theta}$

Problem 3: $\frac{d}{dx}\arctan{\log_{10}{x}}$ is equal to:

(a) $\frac{1}{1 + (\log_{10}{x})^{2}}$ (b) $\frac{1}{x \log_{10}{(1+ (\log_{10}{x})^{2})}}$ (c) $\frac{1}{x(1+(\log_{10}{x})^{2})}$ (d) $\frac{1}{10 \log{x}(1+(\log_{10}{x})^{2})}$

Problem 4: If $\sin^{2}(mx) + \cos^{2}(ny)=a^{2}$, then $\frac{dy}{dx}$ is equal to:

(a) $\frac{m \sin{(2mx)}}{n \sin{(2ny)}}$ (b) $\frac{n\sin{(2mx)}}{m\sin{(2ny)}}$ (c) $\frac{n\sin{(2ny)}}{m\sin{(2mx)}}$ (d) $\frac{-m\sin{(2mx)}}{n\sin{(2ny)}}$

Problem 5: $\frac{d}{dx}(\frac{\tan{x}-\cot{x}}{\tan{x}+\cot{x}})$ is equal to:

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

Problem 6: If $y=\log_{5}{(\log_{5}{x})}$ then the value of $\frac{dy}{dx}$ is

(a) $\frac{1}{x \log_{5}{x}}$ (b) $\frac{1}{x \log_{5}{x}. (\log{5})^{2}}$ (c) $\frac{1}{\log{5}.x\log{x}}$ (d) $\frac{1}{x(\log_{5}{x})^{2}}$

Problem 7: $\frac{d}{dx}(ax+b)^{cx+d}$ is equal to:

(a) $(ax+b)^{cx+d}(\frac{cx+d}{ax+b} + \log{(ax+b)})$ (b) $(ax+b)^{cx+d}(\frac{cx+d}{ax+b} + c \log{(ax+b)})$ (c) $a(ax+b)^{cx+d}$ (d) none

Problem 8: $\frac{d}{dx}(\log{(\frac{\sin{(x-b)}}{\sin{(x-a)}})})$ is equal to:

(a) $\frac{\cos{(a-b)}}{\sin{(x-a)}\sin{(x-b)}}$ (b) $\frac{\sin{(b-a)}}{\sin{(x-a)}\sin{(x-b)}}$

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

Problem 9: If $y = \sqrt{\frac{\sec{x}+\tan{x}}{\sec{x}-\tan{x}}}$ and $0, then $\frac{dy}{dx}$ is :

(a) $\sec{x}(\sec{x}-\tan{x})$ (b) $\sec{x}(\sec{x}+\tan{x})$ (c) $\tan{x}(\sec{x}+\tan{x})$ (d) $\tan{x}(\sec{x}-\tan{x})$

Problem 10: $\frac{d}{dx}e^{ax}(a\sin{(bx)}-b\cos{(bx)})$ is equal to:

(a) $e^{ax}(\sin{(bx)})$ (b) $(a^{2}+b^{2})e^{ax}\sin{(bx)}$ (c) $e^{ax}\cos{(bx)}$ (d) $(a^{2}+b^{2})e^{ax}\cos{(bx)}$

Cheers,

Nalin Pithwa.

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