If $y=x \log \left(\frac{1}{a x}+\frac{1}{a}\right)$,then $x(x+1) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=$

If $y = \sin^2 \alpha + \cos^2(\alpha + \beta) + 2 \sin \alpha \sin \beta \cos(\alpha + \beta)$,then find $\frac{d^3y}{d\alpha^3}$ (keeping $\beta$ as a constant).

If $y = \cos^{-1} x$,find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ alone.

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 :

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 $h(10)$ is $\qquad$

If $y = x^2 e^{mx}$,where $m$ is a constant,then $\frac{d^3y}{dx^3} = $