If $y=e^{4x} \cos 5x$,then $\frac{d^{2}y}{dx^{2}}$ at $x=0$ is

If $x^3 + y^3 = 6$ and $\frac{d^2y}{dx^2} \cdot \frac{d^2x}{dy^2} = \frac{m}{(xy)^n}$, where $m, n \in R$, then find the value of $\frac{m}{n}$.

If $f$ is twice differentiable such that $f''(x) = -f(x)$,$f'(x) = g(x)$,$h'(x) = [f(x)]^2 + [g(x)]^2$,and $h(0) = 2$,$h(1) = 4$,then the equation $y = h(x)$ represents:

If $y=A \sin x+B \cos x$,then prove that $\frac{d^{2} y}{d x^{2}}+y=0$.

If ${I_n} = \frac{d^n}{dx^n}(x^n \log x)$,then ${I_n} - n{I_{n - 1}} = $

If $y = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$,then find the $n^{th}$ derivative $y_n$.