If $\cos ^{-1}\left(\frac{y}{b}\right)=n \log \left(\frac{x}{n}\right)$,then

If $x^{\frac{2}{5}}+y^{\frac{2}{5}}=a^{\frac{2}{5}}$,then $\frac{dy}{dx} = $

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a twice differentiable function. If for some $a \neq 0$,$\int_0^1 f(\lambda x) d\lambda = a f(x)$,$f(1) = 1$ and $f(16) = \frac{1}{8}$,then $16 - f^{\prime}\left(\frac{1}{16}\right)$ is equal to . . . . . .

If $\log _{10}\left(\frac{x^{3}-y^{3}}{x^{3}+y^{3}}\right)=2$,then $\frac{dx}{dy} = $

If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{dy}{dx}=$

If $f(x)$ is a continuous and differentiable function satisfying $f(x) \cdot f(f(x)) = x^2 + 1$,$f(1) = 2$,and $f'(1) = k$,then the value of $f'(2)$ is