If $y=7^x\log(5x^3)$, then $\dfrac{dy}{dx}$ is equal to: (consider $\log_e x = \log x$)

The function $f: \mathbb{R} \to \mathbb{R}$ defined by

$f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$

is

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

Let $f(x) = |3-x| + (3+x)$ be a function, then which of the following statements are TRUE?

A. $f(x)$ is continuous at $x=3$

B. $f(x)$ is continuous at $x=-3$

C. $f(x)$ is differentiable at $x=3$

D. $f(x)$ is differentiable at $x=-3$

Choose the correct answer from the options given below:

If $y = (1+x)(1+x^2)(1+x^4)$, then $\frac{dy}{dx}$ at $x = -1$ is

If $y = e^{acos^{-1}x}, -1 < x < 1$, then $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$ is equal to

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is