Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ =

If $y=e^{2\log_e t}$ and $x=\log_e e^{t^2}$, then the value of $\frac{d^2y}{dx^2}$ is

If the function $f(x)=\begin{cases}\frac{1-\cos kx}{\sin^2 x}, & x\ne0\\\frac{1}{2}, & x=0\end{cases}$ is continuous at $x=0$, then the value of $k$ is

If $x = 3t^2$ and $y = \sqrt{t}$, then the value of $\frac{d^2y}{dx^2}$ is

Which of the following functions $f(x)$ are differentiable at $x = 0$?

(A) $|x|$

(B) $|x - 1|$

(C) $|\sin x|$

(D) $|\cos x|$

(E) $x^2$

Choose the correct answer from the options given below:

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.