Differentiation

Let $f(x)$ be a second degree polynomial. If $f(1)=f(-1)$ and $p, q, r$ are in A.P., then $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are

If ' $f$ ' is the inverse function of ' $g$ ' and $g^{\prime}(x)=\frac{1}{1+x^n}$, then the value of $f^{\prime}(x)$ is

\(\text { If } y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \cdot \log _e x}\right] \text {, then } \frac{d^2 y}{d x^2}=\)

If \(\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)\) denotes the \(\mathrm{n}^{\text {th }}\) derivative \((\mathrm{n}=1,2)\) of \(\mathrm{U}(x)=\frac{\mathrm{L} x+\mathrm{M}}{x^2-2 \mathrm{~B} x+\mathrm{C}}\) (L, M, B, C are constants), then $$\...

Let \(f(x) = {x^m}\), m being a non-negative integer. The value of m so that the equality \(f'(a + b) = f'(a) + f'(b)\) is valid for all a, b > 0 is

If \(x = \sin \theta\) and \(y = \sin k\theta\), then \((1 - {x^2}){y_2} - x{y_1} - \alpha y = 0\), for \(\alpha=\)

If \(y = {\log ^n}x\), where \({\log ^n}\) means \({\log _e}{\log _e}{\log _e}\,...\) (repeated n times), then \(x\log x{\log ^2}x{\log ^3}x\,.....\,{\log ^{n - 1}}x{\log ^n}x{{dy} \over {dx}}\) is equal to

The function \(y = {e^{kx}}\) satisfies \(\left( {{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}}} \right)\left( {{{dy} \over {dx}} - y} \right) = y{{dy} \over {dx}}\). It is valid for

Let \({\cos ^{ - 1}}\left( {{y \over b}} \right) = {\log _e}{\left( {{x \over n}} \right)^n}\), then \(A{y_2} + B{y_1} + Cy = 0\) is possible for, where \({y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}\)

Suppose \(f:R \to R\) be given by \(f(x) = \left\{ \matrix{ 1,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,x = 1 \hfill \cr {e^{({x^{10}} - 1)}} + {(x - 1)^2}\sin {1 \over {x - 1}},\,\mathrm{if}\,x \ne 1 \hfill \cr} \right.\)
then

If \(y = {e^{{{\tan }^{ - 1}}x}}\), then

A bulb is placed at the centre of a circular track of radius 10 m. A vertical wall is erected touching the track at a point P. A man is running along the track with a speed of 10 m/sec. Starting from P the speed with which his shadow is run...

Let \(g(x) = \int\limits_x^{2x} {{{f(t)} \over t}dt}\) where x > 0 and f be continuous function and f(2x) = f(x), then

Let \(y = {{{x^2}} \over {{{(x + 1)}^2}(x + 2)}}\). Then \({{{d^2}y} \over {d{x^2}}}\) is

Let \(f(x) = {x^4} - 4{x^3} + 4{x^2} + c,\,c \in R\). Then

Let f(x) > 0 for all x and f'(x) exists for all x. If f is the inverse function of h and \({h'(x) = {1 \over {1 + \log x}}}\). Then, f'(x) will be

Let f and g be differentiable on the interval I and let a, b \(\in\) I, a < b. Then,

Let f(x) be a derivable function, f'(x) > f(x) and f(0) = 0. Then,

Let \({f_1}(x) = {e^x}\), \({f_2}(x) = {e^{{f_1}(x)}}\), ......, \({f_{n + 1}}(x) = {e^{{f_n}(x)}}\) for all n \(\ge\) 1. Then for any fixed n, \({d \over {dx}}{f_n}(x)\) is

The equation x log x = 3 \(-\) x

If f(x) = xn, being a non-negative integer, then the values of n for which f'(\(\alpha\) + \(\beta\)) = f'(\(\alpha\)) + f'(\(\beta\)) for all \(\alpha\), \(\beta\) > 0 is

If \(f(x) = {\log _5}{\log _3}x\), then f'(e) is equal to

Let \(f(x) = {x^3}{e^{ - 3x}},\,x > 0\). Then the maximum value of f(x) is

The approximate value of \(\root 5 \of {33}\) correct to 4 decimal places is

If \(y = 2{x^3} - 2{x^2} + 3x - 5\), then for x = 2 and \(\Delta\)x = 0.1 the value of \(\Delta\)y is

If \(y = {\tan ^{ - 1}}{{\sqrt {1 + {x^2}} - 1} \over x}\), then y'(1) =

Let \(f(x) = ta{n^{ - 1}}x\). Then \(f'(x) + f''(x)=0\), when x is equal to

If \(y = {A \over x} + B{x^2}\), then \({x^2}{{{d^2}y} \over {d{x^2}}}\) =

If x2 + y2 = 4, then \(y{{dy} \over {dx}} + x =\)

If \(y = {\tan ^{ - 1}}\sqrt {{{1 - \sin x} \over {1 + \sin x}}}\), then the value of \({{dy} \over {dx}}\) at \(x = {\pi \over 6}\) is

The second order derivative of a sin3t with respect to a cos3t at \(t = {\pi \over 4}\) is

The function \(f(x) = {e^{ax}} + {e^{ - ax}},a > 0\) is monotonically increasing for

Select the correct statement from (a), (b), (c), (d). The function \(f(x) = x{e^{1 - x}}\)

The value of \({{dy} \over {dx}}\) at \(x = {\pi \over 2}\), where y is given by \(y = {x^{\sin x}} + \sqrt x\) is

If \(x = {e^t}\sin t\), \(y = {e^t}\cos t\) then \({{{d^2}y} \over {d{x^2}}}\) at x = \(\pi\) is