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?

If $x$ and $y$ are connected parametrically by the equations, without eliminating the parameter, Find $\frac{dy}{dx}$. $x = a(\theta - \sin \theta), y = a(1 + \cos \theta)$

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

Differentiate the following w.r.t. $x$: $\cos(\log x + e^x), x > 0$

Differentiate the following w.r.t. $x$: $e^{x^3}$

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 $\frac{dy}{dx}$. $x = a \cos \theta, y = b \cos \theta$

Find all the points of discontinuity of $f$ defined by $f(x) = |x| - |x + 1|$.

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

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

Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx^2, & \text{if } x \le 2 \\ 3, & \text{if } x > 2 \end{cases}$ at $x = 2$

Examine the following functions for continuity. (d) $f(x) = |x - 5|$

Differentiate the functions with respect to $x$. $2\sqrt{\cot(x^2)}$

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

If $x$ and $y$ are connected parametrically by the equations, without eliminating the parameter, Find $\frac{dy}{dx}$. $x = \frac{\sin^3 t}{\sqrt{\cos 2t}}, y = \frac{\cos^3 t}{\sqrt{\cos 2t}}$

Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & \text{if } x \neq \frac{\pi}{2} \\ 3, & \text{if } x = \frac{\pi}{2} \end{cases}$ at $x = \frac{\pi}{2}$

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

Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}$

Differentiate the following w.r.t. $x$: $\frac{\cos x}{\log x}, x > 0$