If $y = x + e^x$,then $\frac{d^2x}{dy^2}$ is:

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

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:

The second order derivative of $\cos^{-1}(4x^3 - 3x)$ with respect to $\cos^{-1}(2x^2 - 1)$, where $\frac{1}{2} < x < 1$ is

If $\sqrt{r} = a e^{\theta \cot \alpha}$ where $a$ and $\alpha$ are real numbers,then $\frac{d^{2} r}{d \theta^{2}} - 4 r \cot^{2} \alpha$ 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