If $y = \log_e \sin(e^x + 5x + 500)$ then $\dfrac{dy}{dx}$ is equal to

If $x = a\left(t - \dfrac{1}{t}\right)$, $y = b\left(t + \dfrac{1}{t}\right)$, then $\dfrac{dy}{dx} =$

Observe the graph of a function $y = f(x)$ and identify the correct statements (s):

A. Function is continuous but not differentiable at $x = a$

B. Function is not defined at $x = a$

C. Function is defined at $x = a$ but not differentiable.

D. Given that at $x = a, y = b$.

Choose the correct answer from the options given below:

A function $f(x)$ is defined by :

$f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$

Which of the following is true ?

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

If $x = e^{y + e^{y + e^{y + \ldots \infty}}}$, $x > 0$, then $\frac{dy}{dx}$ is equal to

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is