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

Find the second-order derivative of the function $f(x) = x^{3} \log x$.

Let $f(x)=ax^3+bx^2+cx+41$ be such that $f(1)=40, f'(1)=2$ and $f''(1)=4$. Then $a^2+b^2+c^2$ is equal to :

$A$ function $y = f(x)$ has a second-order derivative $f''(x) = 6(x - 1)$. If its graph passes through the point $(2, 1)$ and at that point the tangent to the graph is $y = 3x - 5$,then the function is

$y=e^{a \sin ^{-1} x} \Rightarrow (1-x^2) y_{n+2}-(2 n+1) x y_{n+1}$ is equal to

Eliminating $a$ and $b$ from the relation $y = a \log x + b$,we get $......$