Let $f(x)=x^5+2x^3+3x+1$,$x \in R$,and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in R$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to:

If ${y^x} + {x^y} = {a^b}$,then $\frac{dy}{dx} = $

Find $\frac{dx}{dy}$ for the equation $\sin^{2} y + \cos(xy) = \pi$.

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \,d t=\int_{0}^{x} f(t) \,d t$ for $0 \leq x \leq 1$ and $f(0)=0$,then $\lim_{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) \,d t$ is:

If $y=x \tan y$,then $\frac{d y}{d x}=$

If $x=e^{(x/y)}$,then $\frac{dy}{dx}=$