Calculus

What is the value of $\lim_{x \to 0} \frac{e^x - 1}{x}$?

Find $\frac{d}{dx}(\frac{x}{2})$.

The derivative of $\ln(x^3)$ is:

If $y = \sin(x^2)$, then $\frac{dy}{dx}$ is:

If $y = a e^{mx} + b e^{-mx}$, find the value of $y'' - m^2 y$.

Find $\frac{dy}{dx}$ if $x^2 + xy + y^2 = 100$.

Let $f(x) = x \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Is $f(x)$ continuous and/or differentiable at $x=0$?

If $f(x) = \begin{cases} \frac{1-\cos kx}{x \sin x} & x \neq 0 \\ 1/2 & x = 0 \end{cases}$ is continuous at $x = 0$, then $k$ is:

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

If $f(x) = |x|^3$, then $f''(0)$ is:

Differentiate $y = \sin(x^3)$ with respect to $x$.

Find $\frac{dy}{dx}$ if $y = \log_{10} x$.

If $x - y = \pi$, then $\frac{dy}{dx}$ is equal to:

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

Consider the implicit function $x^2 + y^2 = 25$ representing a circle centered at the origin. If a point $P(x, y)$ moves along the circle such that $x = 3$ and $y = 4$, find the value of the derivative $\frac{dy}{dx}$ at this point.

Given the function $y = e^{x \sin x}$, find the value of the derivative $\frac{dy}{dx}$ at the point where $x = \pi$. The graph of the function $f(x) = e^x$ is shown for reference.

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

A signal processor uses a filter with a response curve $y = \sin(x^2 + \ln x)$. Find the derivative $dy/dx$ that characterizes the signal's rate of change at $x = 1$.

The profile of a high-speed cam is given by $f(x) = \frac{e^{2x} - 1}{e^{2x} + 1}$. A technician needs to find the steepness of the cam at the origin ($x=0$). Calculate $f'(0)$.

Two mechanical linkages are connected such that their displacement follows $x^y = e^{x-y}$. Find the gradient of the displacement $dy/dx$ at the point $(1, 1)$.

If $x = 2t$ and $y = 2/t$, then the value of $\frac{dy}{dx}$ is:

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

If $x = a \cos t$ and $y = b \cos t$, then find $\frac{dy}{dx}$.

If $y = (x+1)^{(x+1)}$, find the value of $\frac{dy}{dx}$ at $x=0$.

If $y = (1+x)(1+x^2)(1+x^4)(1+x^8)$, find $\frac{dy}{dx}$ at $x=1$.

Find $\frac{d^2y}{dx^2}$ at $\theta = \pi/2$ if $x = a(\theta - \sin \theta)$ and $y = a(1 - \cos \theta)$.

If $y = (x+1)(x+2)(x+3)$, the value of $\frac{dy}{dx}$ at $x=0$ using logs is:

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

If $y = (\tan x)^{\tan x^{\dots \infty}}$, then $\frac{dy}{dx}$ at $x = \frac{\pi}{4}$ is:

If $x = a(\cos 2t(1 + \cos 2t))$ and $y = a(\sin 2t(1 - \cos 2t))$, find $\frac{dy}{dx}$ at $t = \frac{\pi}{8}$.

Find the second order derivative of $y = \log(ax+b)$.

If $y = e^x \cos x$, then $\frac{d^2y}{dx^2}$ is:

If $y = e^x \sin x$, then $y''$ is:

Find the second order derivative of $y = \frac{x^3}{3} + \frac{x^2}{2} + x$.

If $y = \log(\sin x)$, find the value of $y'' + (y')^2$.

If $y = \sin x^2$, find $\frac{d^2y}{dx^2}$.

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

If $x = a \sec^3 \theta$ and $y = a \tan^3 \theta$, find the value of $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{4}$.

If $y = x^2 e^x$, then $y_2$ is equal to:

If $x = a(t + \frac{1}{t})$ and $y = a(t - \frac{1}{t})$, find $\frac{d^2y}{dx^2}$.

A curve is defined by the function $f(x) = x^2 - 4x + 5$. At what point on the curve is the tangent line horizontal?

Find the local minimum value of the function $f(x) = x^2 + \frac{16}{x}$ for $x > 0$.

If the side of a square is increasing at the rate of $0.2$ cm/s, find the rate of increase of its perimeter.

Find the point on the curve $y = x^2 - 2x + 3$ where the tangent is horizontal.

A cylindrical can is to be made to hold $1$ liter ($1000$ cm$^3$) of oil. Find the dimensions (radius $r$) which will minimize the cost of the metal to make the can.

Find the point on the curve $y = x^3 - 11x + 5$ at which the tangent is $y = x - 11$.

The curve $y = ax^3 + bx^2 + cx + d$ has a local maximum at $(0, 4)$ and a local minimum at $(2, 0)$. Find the value of $a$.

The angle $\theta$ between two equal sides of an isosceles triangle with fixed side length $a$ is increasing at a rate of 2 radians per second. At what rate is the area of the triangle changing when $\theta = \pi/3$?

Determine the maximum value of the function $f(x) = \sin x + \cos x$ on the interval $[0, \pi]$.

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the piece used for the square so that the combined area of the square and the circle is minimum?

Find $\int e^{x/3} dx$.

Evaluate $\int \cos(x/2) dx$.

Using partial fractions, evaluate $\int \frac{dx}{(x-3)(x-4)}$.

Evaluate $\int x e^{3x} dx$ by parts.

Solve $\int \frac{\ln(\tan x)}{\sin x \cos x} dx$ using substitution.

Find the general formula for $\int \frac{1}{x^2-a^2} dx$ using partial fractions.

Evaluate $\int \frac{x^5}{1+x^{12}} dx$.

Evaluate $\int \frac{x^2 - 1}{x^4 + x^2 + 1} dx$.

Find the integral: $\int \frac{x^2 + 1}{x^4 + x^2 + 1} dx$.

Evaluate $\int \frac{dx}{x^3 + x^4}$.

Evaluate the definite integral $\int_{0}^{\pi} \cos(\frac{x}{2}) dx$.

Find the value of $\int_{1}^{2} e^{x} dx$.

Evaluate $\int_{0}^{1} x^{1/3} dx$.

The integral $\int_{-\pi}^{\pi} \sin x dx$ is equal to:

Evaluate the definite integral: $\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx$

The value of $\int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}}$ is:

Evaluate $\int_{0}^{1} \frac{dx}{e^x + e^{-x}}$.

Evaluate the definite integral: $I = \int_{0}^{\pi/2} \frac{x}{\sin x + \cos x} dx$

What is the value of $\int_{0}^{\pi} \frac{x \tan x}{\sec x + \tan x} dx$?

Evaluate $I = \int_{0}^{1} \frac{\ln x}{\sqrt{1-x^2}} dx$.

Find the value of $\int_0^1 (1 - x^2) dx$.

Evaluate the integral $\int_1^4 \frac{2}{\sqrt{x}} dx$.

Calculate $\frac{d}{dx} \int_x^5 \cos t dt$.

Find the value of $\int_0^{\pi/3} \sec x \tan x dx$.

Evaluate the definite integral $\int_0^1 \frac{1}{x^2 + 3x + 2} dx$.