Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$, then g'(63) is equal to ________________.

Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b $\in$ R, g'(a) = 5 and g(a) = b,…

If $y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x$, then $96…

Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1. If $${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ -…

If y = $$\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}} $$, then ${{dy} \over {dx}}$…

Let $$f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$$. Then $f\left(\frac{7 \pi}{12}\right)…