Find the second order derivatives of the functions $x^2 + 3x + 2$

Find $\frac{dy}{dx}$ in the following: $\sin^2 y + \cos xy = \kappa$

Examine the following functions for continuity. (b) $f(x) = \frac{1}{x-5}$, $x \neq 5$

For what value of $\lambda$ is the function defined by $f(x) = \begin{cases} \lambda(x^2 - 2x), & \text{if } x \le 0 \\ 4x + 1, & \text{if } x > 0 \end{cases}$ continuous at $x = 0$? What about continuity at $x = 1$?

Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx + 1, & \text{if } x \le \pi \\ \cos x, & \text{if } x > \pi \end{cases}$ at $x = \pi$

Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx + 1, & \text{if } x \le 5 \\ 3x - 5, & \text{if } x > 5 \end{cases}$ at $x = 5$

If $u, v$ and $w$ are functions of $x$, then show that $\frac{d}{dx} (u \cdot v \cdot w) = \frac{du}{dx} v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \frac{dw}{dx}$ in two ways - first by repeated application of product rule, second by logarithmic differentiation.

Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$

Differentiate the following w.r.t. $x$: $e^x + e^{x^2} + ... + e^{x^5}$

Differentiate the functions with respect to $x$. $\sin(x^2 + 5)$

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

Find the second order derivatives of the functions. $\tan^{-1} x$

Differentiate the functions with respect to $x$. $\cos(\sin x)$

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

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.

Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (iii) by logarithmic differentiation. Do they all give the same answer?

Find the second order derivatives of the functions. $e^{6x} \cos 3x$

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

Differentiate the functions w.r.t. $x$. $(\log x)^x + x^{\log x}$

Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1}(2x\sqrt{1-x^2}), -\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$