Differentiation - Calculus PYQ | Mathematics Exams

\(\frac{d}{d x}\left[\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\right]\) is equal to

If \(\log \left(\sqrt{1+x^2}-x\right)=y\left(\sqrt{1+x^2}\right)\), then \(\left(1+x^2\right) \frac{d y}{d x}+x y\) is equal to

If \(y=e^{x^2+e^{x^2+e^{x^2+\cdots \infty}}}\), then \(\frac{d y}{d x}\) is equal to

If \(x^2+y^2=1\), then

If \(y=x+\frac{1}{x}\), then which among the following holds?

\(f(x)=\sqrt{x^2+1}: g(x)=\frac{x+1}{x^2+1}: h(x)=2 x-3\), then the value of \(f^{\prime}\left[h^{\prime}\left(g^{\prime}(x)\right)\right]\) is equal to

For which value(s) of \(a\) \(f(x)=-x^3+4 a x^2+2 x-5\) is decreasing for every \(x\) ?

If \(3 \sin x y+4 \cos x y=5\), then \(\frac{d y}{d x}\) is equal to

If \(x=\sec \theta-\cos \theta\) and \(y=\sec ^n \theta-\cos ^n \theta\), then \(\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2\) is equal to

If \(y=\log _{\cot x} \tan x-\log _{\tan x} \cot x +\tan ^{-1}\left(\frac{4 x}{4-x^2}\right)\), then \(\frac{d y}{d x}\) is equal to

If \(y=\left(1+\frac{1}{x}\right)\left(1+\frac{2}{x}\right)\left(1+\frac{3}{x}\right) \ldots\left(1+\frac{n}{x}\right)\) and \(x \neq 0\). When \(x=-1, \frac{d y}{d x}\) is equal to

If \(f(x)=2x^2+3x-5\), then the value of \(f'(0)+3f'(-1)\) is equal to

If \(x \neq 0\) and \(f(x)\) satisfies \(8 f(x)+6 f(1 / x) =x+5\), then \(\frac{d}{d x}\left(x^2 f(x)\right)\) at \(x=1\) is

If \(f(x)=\cot ^{-1}\left(\frac{x^x+x^{-x}}{2}\right)\), then \(f^{\prime}(1)=\)

\(y=x^3-a x^2+48 x+7\) is an increasing function for all real values of \(x\), then \(a\) lies in the interval

If \(x=f(\theta)\) and \(y=g(\theta)\), then \(\frac{d^2 y}{d x^2}=\)

Assertion (A) \(\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}\left(\cot x+\frac{2}{x}-\frac{1}{x \log x}\right)\)

Reason (R) \(\frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u v}{w}\left[\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}+\frac{w^{\prime}}{w}\right]\)