Continuity and Differentiability

If $y = e^x + e^{-x}$, then $\frac{dy}{dx}$ is:

The derivative of $\text{cosec } x$ with respect to $x$ is:

The derivative of $\log(\log x)$ with respect to $x$ is:

What is the derivative of $\cos(\sin x)$ with respect to $x$?

If $y = \tan^{-1}\left(\frac{\cos x + \sin x}{\cos x - \sin x}\right)$, then $\frac{dy}{dx}$ is:

If $y = \log \sqrt{\frac{1-\cos x}{1+\cos x}}$, then $\frac{dy}{dx}$ is:

If $f(x) = \begin{cases} 1, & x \leq 3 \\ ax+b, & 3 < x < 5 \\ 7, & x \geq 5 \end{cases}$ is continuous, then the values of $a$ and $b$ are:

If $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x} & x \neq \pi/2 \\ 3 & x = \pi/2 \end{cases}$ is continuous at $x = \pi/2$, then $k$ is:

The value of $c$ in Rolle's Theorem for $f(x) = e^x (\sin x - \cos x)$ in $[\pi/4, 5\pi/4]$ is:

If $y = e^x \sin x$, then $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y$ is:

Differentiate $y = \sec(\tan \sqrt{x})$. This is a triple composition. What is the first step of the chain rule derivative?

Find the derivative of $y = 5^{x^2}$.

If $y = \log(x^2 + 2x + 1)$, find $\frac{dy}{dx}$.

Differentiate $y = \cos(\sin x)$.

If $y = (x^2 + 1)^{1/x}$, this function requires logarithmic differentiation which is a form of the chain rule. What is the derivative of $\ln y$?

Differentiate $y = \log(\sec x + \tan x)$.

If $y = e^{\sqrt{3x^2 - 6x + 5}}$, what is the value of $\frac{dy}{dx}$ at $x = 1$?

Given the function $y = \sqrt{a^2 - x^2}$, representing the upper semi-circle shown in the figure, calculate the derivative $\frac{dy}{dx}$ using the chain rule and evaluate it at $x = \frac{a}{2}$.

A chemical reaction rate is modeled by $R(x) = \frac{1}{1 + e^{-x^2}}$. A chemist wants to find the sensitivity of the rate at $x = 1$. Find $R'(1)$.

An object's position is given by $s(t) = e^{\sin(t^2)}$. Find the velocity of the object $v(t) = \frac{ds}{dt}$ at $t = \sqrt{\pi}$.

The derivative of $\log(\log x)$ with respect to $x$ is:

What is the derivative of $y = \log_{10} x$?

Find $\frac{dy}{dx}$ if $x - y = \pi$.

Differentiate $y = \sqrt{e^x}$ with respect to $x$.

Given the implicit function $x^3 + y^3 = 3axy$, find the slope of the tangent to the curve at the point $(\frac{3a}{2}, \frac{3a}{2})$. The diagram shows the Folium of Descartes which represents this equation.

If $x^y = e^{x-y}$, find the expression for $\frac{dy}{dx}$. The provided diagram illustrates the logarithmic growth curve $y = \ln(x)$.

Find the derivative of $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to $x$ for $|x| < 1$. The diagram shows the plot of the function within the specified interval.

A sound wave is described by $y = e^{\sin(x^2)}$. To find the intensity gradient, compute $\frac{dy}{dx}$ at $x = \sqrt{\frac{\pi}{2}}$.

A specialized sensor outputs a signal $S = \sin^{-1}(\frac{x^2-1}{x^2+1})$. A processor calculates the derivative of $S$ with respect to $x$. What is $\frac{dS}{dx}$ at $x = 1$?

In a study of fluid dynamics, the velocity potential is given by $x^3 + y^3 = 3axy$. Determine the rate of change of $y$ with respect to $x$ at the point $(\frac{3a}{2}, \frac{3a}{2})$.

If $y = x^{1-x}$, find $\frac{dy}{dx}$.

If $y = (1-x)^x$, find $\frac{dy}{dx}$.

If $y = x^7$, find $\frac{dy}{dx}$ (can be verified by logs).

The derivative of $y = 7^x$ is:

Find $\frac{dy}{dx}$ if $y = \cos(x^x)$.

If $y = (\tan x)^{\log x}$, then $\frac{dy}{dx}$ is:

Find the derivative of $y = (x \cos x)^x$.

If $y = (\ln x)^{x^x}$, then the value of the derivative $\frac{dy}{dx}$ at $x = e$ is:

Find the derivative of the function $y = (\tan x)^{\ln(\sin x)}$ with respect to $x$.

If $y = (\sin x + \cos x)^{1 + \tan x}$, find the value of $\frac{dy}{dx}$ at $x = \frac{\pi}{4}$.

The coordinates of a moving point are given by $x = e^t$ and $y = e^{2t}$. The graph represents the trajectory of the point in the first quadrant. Find the value of $\frac{dy}{dx}$ expressed in terms of $t$.

Consider the curve defined parametrically by $x = 2t^2$ and $y = 4t$. The graph of this parabola is shown below. Determine the slope of the tangent to the curve at the point where $t = 1$.

A particle moves in the $xy$-plane such that its coordinates at any time $t$ are given by $x = a \cos(t)$ and $y = a \sin(t)$, where $a$ is a constant. The path of the particle is a circle as shown in the diagram. Find $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$.

Find $\frac{dy}{dx}$ at $t = \frac{\pi}{2}$ if $x = a(t + \sin t)$ and $y = a(1 + \cos t)$.

If $x = t^2$ and $y = t^3$, find $\frac{d^2y}{dx^2}$ in terms of $t$.

If $x = a \cos \theta$ and $y = b \sin \theta$, find $\frac{d^2y}{dx^2}$.

If $x = a \sin 2t (1 + \cos 2t)$ and $y = a \cos 2t (1 - \cos 2t)$, find $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$.

A camera tracks an object moving along a path given by $x = 2t^2$ and $y = t^4 - 2t^2$. The camera's software must calculate the rate at which the slope changes with respect to horizontal distance. Determine $\frac{d^2y}{dx^2}$ at $t = 2$.

A project involves calculating the curvature of a cable modeled by $x = c \ln(\sec \psi + \tan \psi)$ and $y = c \sec \psi$. If the slope of the tangent is $\frac{dy}{dx}$, find the value of $\frac{d^2y}{dx^2}$ when $\psi = \frac{\pi}{4}$.

The displacement of a buoy in a wave tank is described by the parametric equations $x = a(1 + \cos \phi)$ and $y = a(\phi + \sin \phi)$. An engineer needs to find the concavity of the motion path. Find $\frac{d^2y}{dx^2}$ in terms of $\phi$.

Find the second order derivative of $y = x^2 \log x$.

If $y = \log(\sin x)$, find $y''$.

Find $\frac{d^2y}{dx^2}$ for $y = \frac{1}{x^3}$.

Calculate the second order derivative of $y = \sin(3x+2)$.

If $y = \log(\tan(x/2))$, then find $y''$.

Given $y = x^{10}$, find $\frac{d^2y}{dx^2}$.

If $y = \cos(\sin x)$, find $y''$.

If $y = \log(x^2 e^x)$, then the value of $y''$ is:

If $y = 3e^{2x} + 2e^{3x}$, find $y'' - 5y' + 6y$.

If $y = \log(\tan(\frac{\pi}{4} + \frac{x}{2}))$, find the value of $y''$ at $x = 0$.

At $x = 3$, the function $f(x) = \begin{cases} x + 3, & x < 3 \\ 6, & x = 3 \\ 2x, & x > 3 \end{cases}$ is:

Which of the following is always true for a continuous function $f$ on $[a, b]$?

If $f(x) = \begin{cases} \frac{x^2 - 3x + 2}{x - 2}, & x \neq 2 \\ k, & x = 2 \end{cases}$ is continuous at $x = 2$, then $k$ is:

The function $f(x) = |x-2|$ is continuous at:

For what value of $k$ is $f(x) = \begin{cases} \frac{\sin(3x^2)}{x^2}, & x \neq 0 \\ k, & x = 0 \end{cases}$ continuous at $x = 0$?

Find $k$ such that $f(x) = \begin{cases} \frac{\sin(\pi \cos^2 x)}{x^2}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$.

If $f(x) = \begin{cases} \frac{10^x - 2^x - 5^x + 1}{x^2}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$, then $k$ is:

Identify the value of $k$ so that the function $f(x) = \begin{cases} \frac{8^x - 2^x}{k^x - 1} & x \neq 0 \\ 2 & x = 0 \end{cases}$ is continuous at $x=0$ (given $k>0, k \neq 1$ and $k$ is the base). Wait, let's rephrase: if $f(x) = \frac{8^x - 2^x}{x}$ for $x \neq 0$ and $f(0) = \ln k$, find $k$.