If $y = Ae^{mx} + Be^{nx}$, show that $\frac{d^2y}{dx^2} - (m + n)\frac{dy}{dx} + mny = 0$

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 functions w.r.t. $x$. $x^{x \cos x} + \frac{x^2+1}{x^2-1}$

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

Is the function defined by $f(x) = \begin{cases} x + 5, & \text{if } x \le 1 \\ x - 5, & \text{if } x > 1 \end{cases}$ a continuous function?

Differentiate the following w.r.t. $x$: $\sin(\tan^{-1} e^{-x})$

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

Is the function $f$ defined by $f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases}$ continuous at $x = 0$? At $x = 1$? At $x = 2$?

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 $y = 3 \cos(\log x) + 4 \sin(\log x)$, show that $x^2 y_2 + xy_1 + y = 0$

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 $\frac{dy}{dx}$. $x = a\left(\cos t + \log \tan\left(\frac{t}{2}\right)\right), y = a \sin t$

Differentiate the functions w.r.t. $x$. $x^x - 2^{\sin x}$

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

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

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 $x = \pi$?

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

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

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