Let $f(x)$ be a twice differentiable function such that $f''(x) = -f(x)$, $f'(x) = g(x)$, and $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. If $h(5) = 11$, then $h(10)$ is equal to ...

If $f(x) = \sin(\sin x)$ and $f''(x) + \tan x f'(x) + g(x) = 0$,then $g(x)$ is

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$, $f^{\prime}(1)=4$, $g^{\prime}(1)=6$, $f(2)=3$, and $g(2)=9$, then $f(1)-g(1)$ is

If $y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \log _e x}\right]$, then $\frac{d^2 y}{d x^2}=$

If $f(x) = |x|^3$,show that $f''(x)$ exists for all real $x$,and find it.

If $x = A \cos 4t + B \sin 4t$,then $\frac{d^2x}{dt^2} = $