## 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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