Derivatives: part 5: IITJEE maths tutorial problems for practice

Problem 1:

The derivative of arcsec (\frac{1}{1-2x^{2}}) w.r.t. \sqrt{1-x^{2}} at x=\frac{1}{2} is

(a) 2 (b) -4 (c) 1 (d) -2

Problem 2:

If y = \sin{\sin{x}} and \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} \tan{x} + f(x)=0, then f(x) =

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

Problem 3:

If f(x) = \log_{a}{\log_{a}{x}}, then f^{'}(x) is

(a) \frac{\log_{a}{e}}{x \log_{e}{x}} (b) \frac{\log_{e}{a}}{x} (c) \frac{\log_{e}{a}}{x\log_{a}{x}} (d) \frac{x}{\log_{e}{a}}

Problem 4:

If y=\log {\tan{\frac{x}{2}}} + \arcsin{\cos{x}}, then \frac{dy}{dx} is

(a) cosec (x) -1 (b) cosec (x) +1 (c) cosec (x) (d) x

Problem 5:

If y^{x}=x^{y}, then \frac{dy}{dx} is

(a) \frac{y}{x} (b) \frac{x}{y} (c) \frac{y(x\log{y}-y)}{x(y\log{x}-x)} (d) \frac{x \log{y}}{y \log{x}}

Problem 6:

Let f, g, h and k be differentiable in (a,b), if F is defined as F(x) = \left | \begin{array}{cc} f(x) & g(x) \\ h(x) & k(x) \end{array} \right | for all a, b, then F^{'} is given by:

(i) \left | \begin{array}{cc} f & g \\ h & k \end{array} \right| + \left | \begin{array}{cc}f & g \\ h^{'}  & k \end{array} \right |

(ii) \left | \begin{array}{cc}f & g^{'} \\ h & k^{'} \end{array}\right | + \left | \begin{array}{cc} f^{'} & g \\ h & k^{'} \end{array} \right |

(iii) \left | \begin{array}{cc}f^{'} & g^{'} \\ h & k \end{array} \right | + \left | \begin{array}{cc}f & g \\ h^{'} & h^{'} \end{array} \right |

(iv) \left | \begin{array}{cc}f & g \\ h^{'} & k^{'} \end{array} \right | + \left | \begin{array}{cc}f^{'} & g \\h & k \end{array} \right |

Problem 7:

If pv=81, then \frac{dp}{dv} at v=9 is equal to:

(i) 1 (ii) -1 (iii) 2 (iv) 3

Problem 8:

If x^{2}+y^{2}=1, then

(i) yy^{''}-2(y^{'})^{2}+1=0 (ii) yy^{''} - (y^{'})^{2}-1=0 (iii) yy^{''} + (y^{'})^{2} + 1 = 0 (iv) yy^{''} - 2(y^{'})^{2}-1=0

Problem 9:

If y = \arctan{\frac{\sqrt{x}-1}{\sqrt{x}+1}} + \arctan{\frac{\sqrt{x}+1}{\sqrt{x}-1}}, then the value of \frac{dy}{dx} will be

(i) 0 (ii) 1 (iii) -1 (iv) - \frac{1}{2}

Problem 10:

Let f(x) = \left | \begin{array}{ccc} x^{3} & \sin{x} & \cos{x} \\ 0 & -1 & 0 \\ p & p^{2} & p^{3} \end{array} \right |, where p is a constant, then \frac{d^{3}}{dx^{3}}(f(x)) at x=0 is

(a) p (b) p+p^{2} (c) p+p^{3} (d) independent of p

Regards,

Nalin Pithwa

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