If $e^y (x + 1) = 1$, show that $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$

Find the values of $a$ and $b$ such that the function defined by $f(x) = \begin{cases} 5, & \text{if } x \le 2 \\ ax + b, & \text{if } 2 < x < 10 \\ 21, & \text{if } x \ge 10 \end{cases}$ is a continuous function.

Is the function defined by $f(x) = x^2 - \sin x + 5$ continuous at $x = \pi$?

Prove that the function $f(x) = x^n$ is continuous at $x = n$, where $n$ is a positive integer.

Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (i) by using product rule

Differentiate the functions with respect to $x$. $\cos(\sqrt{x})$

Differentiate the following w.r.t. $x$: $\sqrt{e^{\sqrt{x}}}, x > 0$

Find $\frac{dy}{dx}$ in the following: $2x + 3y = \sin y$

If $x$ and $y$ are connected parametrically by the equations, without eliminating the parameter, Find $\frac{dy}{dx}$. $x = a \sec \theta, y = b \tan \theta$

If $y = 500e^{7x} + 600e^{-7x}$, show that $\frac{d^2y}{dx^2} = 49y$

Show that the function defined by $f(x) = \cos(x^2)$ is a continuous function.

Find the second order derivatives of the functions. $\log(\log x)$

Find $\frac{dy}{dx}$ in the following: $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$

Discuss the continuity of the following functions: (b) $f(x) = \sin x - \cos x$

Differentiate the functions w.r.t. $x$. $(x \cos x)^x + (x \sin x)^{\frac{1}{x}}$

Find the second order derivatives of the functions $x^{20}$

If $x$ and $y$ are connected parametrically by the equations, without eliminating the parameter, Find $\frac{dy}{dx}$. $x = 2at^2, y = at^4$

Find $\frac{dy}{dx}$ of the functions $x^y + y^x = 1$

Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$

Prove that the function $f(x) = 5x - 3$ is continuous at $x = 0$, at $x = -3$ and at $x = 5$.