Differentiation - Calculus - Mathematics Previous Year Questions

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

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 \(y = {\log ^n}x\), 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

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 \({y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}\)

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

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

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

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

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 \(\mathrm{PU}_2+\mathrm{QU}_1+\mathrm{RU}=0\), holds for