Differentiation

Let $f(x)=\left|\begin{array}{ccc}\sin 3 x & 1 & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^2 \\ \cos 3 x & -1 & 2\left(\cos ^2 \frac{3 x}{2}-\sin ^2 \frac{3 x}{2}\right) \\ \tan 3 x & 4 & 1+2 \tan 3…

A real valued differentiable function $f$ satisfies $f(x)+f(y)+2 x y=f(x+y) \forall x, y \in R$. Given $f^n(0)=0$, then value of $\int_0^{\frac{\pi}{2}} f(\sin x) d x$ will be

If $f(x)=\sec x-\cot x$, then $f^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

Derivative of $x^x$ is

If $f_n(x)=e^{f_{(n-1)}(x)}$, for all $n \in N$ and $f_0(x)=x$, then $\frac{d}{d x}\left\{f_n(x)\right\}$ is equal to

If $f(x)=\frac{x+\cos x}{x-\cos x}, f^{\prime}\left(\frac{\pi}{2}\right)$ is equal to

If $y=\sqrt{\sin x+\sqrt{\sin x+\ldots . .}}$, then $\frac{d y}{d x}$ is

Derivative of the function $f(x)=\log _5\left(\log _7 x\right), x\gt7$ is

If $2^x+2^y=2^{x+y}$, then $\frac{d y}{d x}$ is equal to

The derivative of $f(x)=3|2+x|$ at the point $x_0=-3$ is

If $u=x^2+y^2$ and $x=s+3 t, y=2 s-t$, then $\frac{d^2 u}{d s^2}$ is equal to

If $x=\sin ^{-1}\left(3 t-4 t^3\right)$ and $y=\cos ^{-1}\left(\sqrt{1-t^2}\right)$ then $\frac{d y}{d x}$, is equal to