Let $f$ and $g$ be differentiable functions on $R$ such that $f \circ g$ is the identity function. If for some $a, b \in R$,$g^{\prime}(a) = 5$ and $g(a) = b$,then $f^{\prime}(b)$ is equal to

The derivative of $\sin \left( \log \left( \frac{x + 3}{x} \right) \right)$ with respect to $x$ is:

Let $f: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If $\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) dt}{e^{x^2}-1}=\alpha$,then $8 \alpha^2$ is equal to:

Evaluate: $\frac{d}{d x}\left(3 \cos \left(\frac{\pi}{6}+x^{\circ}\right)-4 \cos ^3\left(\frac{\pi}{6}+x^{\circ}\right)\right) = $ . . . . . .

For the function $f(x) = x^2 - 6x + 8$ where $2 \le x \le 4$,the value of $x$ for which $f'(x)$ vanishes is:

If $y = f\left( \frac{2x - 1}{x^2 + 1} \right)$ and $f'(x) = \sin(x^2),$ then $\frac{dy}{dx} = $