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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