## Derivatives: part 3: IITJEE maths tutorial problems for practice

Problem 1:

Differential coefficient of $\log[10]{x}$ w.r.t. $\log[x]{10}$ is

(a) $\frac{(\log{x})^{2}}{(\log{10})^{2}}$ (b) $\frac{(\log[x]{10})^{2}}{(\log{10})^{2}}$ (c) $\frac{(\log[10]{x})^{2}}{(\log{10})^{2}}$ (d) $\frac{(\log{10})^{2}}{(\log{x})^{2}}$

Problem 2:

The derivative of an even function is always:

(a) an odd function (b) does not exist (c) an even function (d) can be either even or odd.

Problem 3:

The derivative of $\arcsin{x}$ w.r.t. $\arccos{\sqrt{1-x^{2}}}$ is

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

Problem 4:

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, then $\frac{dy}{dx}$ is

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

Problem 5:

$\frac{d}{dx} \arcsin{2x\sqrt{1-x^{2}}}$ is equal to

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

Problem 6:

If $y=\arctan{\frac{x}{2}}-\arccos{\frac{x}{2}}$, then $\frac{dy}{dx}$ is

(a) $\frac{2}{1+x^{2}}$ (b) $\frac{2}{4+x^{2}}$ (c) $\frac{4}{4+x^{2}}$ (d) $0$

Problem 7:

If $y=\arccos{(\frac{\sqrt{1+\sin{x}}+\sqrt{1-\sin{x}}}{\sqrt{1+\sin{x}}-\sqrt{1-\sin{x}}})}$, then $\frac{dy}{dx}$ is equal to:

(a) $\frac{1}{2}$ (b) $\frac{2}{3}$ (c) $3$ (d) $\frac{3}{2}$

Problem 8:

If $y = \arctan{\frac{4x}{1+5x^{2}}} + \arctan{\frac{2+3x}{3-2x}}$, then $\frac{dy}{dx}$ is

(a) $\frac{1}{1+x^{2}}$ (b) $\frac{5}{1+25x^{2}}$ (c) $1$ (d) $\frac{3}{1+9x^{2}}$

Problem 9:

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

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

Problem 10:

If $y^{2}=p(x)$, a polynomial of degree 3, then $2\frac{d}{dx}(y^{3}\frac{d^{2}y}{dx^{2}})$ is equal to

(a) $p^{'''}(x)+p^{'}(x)$ (b) $p^{''}(x).p^{'''}(x)$ (c) $p^{'''}(x).p(x)$ (d) a constant.

Regards,

Nalin Pithwa.

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