Let $f$ be a twice differentiable function such that $f^{\prime \prime}(x) = -f(x)$,$f^{\prime}(x) = g(x)$,and $h(x) = (f(x))^2 + (g(x))^2$. If $h(5) = 1$,then the value of $h(10)$ is

If $y = (ax + b) \cos x$,then $y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = $

Let $y = \log_8 \left( \frac{1-x^2}{1+x^2} \right)$ for $-1 < x < 1$. Then at $x = \frac{1}{2}$,the value of $225(y' - y'')$ is equal to:

Suppose,$f(x)=e^{-\sqrt{x}}+e^{-\frac{1}{x^2}}$. If $f^{\prime \prime}(x)=\alpha \cdot \frac{e^{-\sqrt{x}}}{x}\left(1+\frac{1}{\sqrt{x}}\right)+\beta \cdot \frac{e^{-\frac{1}{x^2}}}{x^4}\left(3-\frac{2}{x^2}\right)$,then $(\alpha, \beta)=$

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

If the three functions $f(x)$,$g(x)$,and $h(x)$ are such that $h(x) = f(x) \cdot g(x)$ and $f^{\prime}(x) \cdot g^{\prime}(x) = c$,where $c$ is a constant,then $\frac{f^{\prime \prime}(x)}{f(x)} + \frac{g^{\prime \prime}(x)}{g(x)} + \frac{2c}{f(x) \cdot g(x)}$ is equal to: