Calculus

Find $f'(x)$ for $f(x) = 10x$ using the definition of the derivative.

In the limit $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, if the limit exists, the function $f$ is said to be _______ at $x$.

Which of the following is equivalent to $\frac{dy}{dx}$ at $x=a$?

If $f(x) = x^2 + x$, what is $f(x+h) - f(x)$?

Find the derivative of $f(x) = (x+1)^3$ using the limit definition.

For $f(x) = 2x^2 + 4x - 1$, find the derivative from first principles.

Using first principles, find the derivative of $f(x) = \frac{1}{3x}$.

Calculate $\lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h}$ to find the gradient of $f(x) = \sqrt{x}$ at $x=1$.

Which of the following describes the derivative of $f(x) = \sqrt[3]{x}$ at $x=0$ using the limit definition?

Determine the derivative of $f(x) = \frac{x+1}{x-1}$ using first principles.

For the function $y = \sin(x) + \cos(x)$, find the derivative $\frac{dy}{dx}$ at $x = 0$.

Determine the derivative of $f(x) = 7x^4$.

The graph of $f(x) = (x-2)^2$ is shown. Find the x-coordinate of the point where the gradient is zero.

Given $y = 4 e^{2x}$, find $\frac{dy}{dx}$.

The diagram shows a rectangle inscribed under the curve $y = e^{-x^2}$. To find the rate of change of the area, we need the derivative of $A(x) = 2x e^{-x^2}$. Find $A'(x)$.

Given $y = \cos(2x - 1)$, find $\frac{dy}{dx}$.

Find the second derivative $f''(x)$ of the function $f(x) = x^4 - 5x^2$.

A curve has the equation $y = x^2 \cos x$. Find the gradient of the curve at $x = \frac{\pi}{2}$.

Consider the function $g(x) = (x^2 - 3)^4$. Find the value of $g''(1)$.

The volume of water in a tank is $V(t) = 10 \ln(t^2 + t + 1)$, where $t$ is in hours. Find the rate at which water is being added to the tank at $t = 1$.

Find the gradient of the tangent to $y = \ln(x^2)$ at $x = 1$.

Find the gradient of the tangent to $y = (3x - 2)^2$ at $x = 1$.

Find the gradient of the normal to $y = x^2 - 5x$ at $x = 2$.

Find the equation of the tangent to $y = x^2 + x$ at the point $(0, 0)$.

Find the equation of the tangent to the curve $f(x) = 2\sin x \cos x$ at $x = 0$.

Find the equation of the tangent line to $y = \ln(x^2 + 1)$ at the point where $x = 1$.

Find the gradient of the normal to the curve $y = e^{3x}$ at the point where $x = 0$.

Find the equation of the tangent line to the curve $y = \ln(x^2)$ at the point where $x = e$.

The curve $y = \frac{1}{x} + \ln(x)$ has a tangent at $x = 1$. This tangent passes through point $(k, 5)$. Find $k$.

What is the equation of the tangent to the curve $f(x) = \cos^2(x) - \sin^2(x)$ at $x = \frac{\pi}{4}$?

Given the function $f(x) = e^{\cos x}$, find the derivative $f'(x)$.

Find the derivative of $y = e^{4x} \sin x$.

Differentiate $f(x) = \frac{4x}{x^2 + 1}$.

Find $\frac{dy}{dx}$ if $y = x^2 \tan x$.

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

Find the derivative of $f(x) = \ln\left(\frac{x}{x+1}\right)$.

Find $f'(x)$ for $f(x) = x^2 \sqrt{x+1}$.

If $y = e^{-2x^2}$, find the second derivative $\frac{d^2y}{dx^2}$.

Find the derivative of $f(x) = (1 - x^3)^{-\frac{1}{2}}$.

Determine the derivative of $f(x) = \ln(x^2 + e^x)$.

At which point does the function $f(x) = (x+3)^2 - 5$ reach its minimum?

Given $f(x) = x^2 + 4$, what is the minimum value of the function?

Find the $x$-coordinate of the stationary point of $f(x) = 8x - x^2$.

On the interval $[0, 2\pi]$, where is $f(x) = \cos(x)$ decreasing?

Given $f'(x) = \frac{x-3}{x}$, where is $f(x)$ increasing for $x > 0$?

Find the $x$-coordinate of the local minimum of $f(x) = x^2 - 4x + 3$.

The function $f(x) = \frac{1}{x^2 + 1}$ has a maximum value. Find this value.

Find the $x$-coordinate of the local maximum of $f(x) = x \ln(x^2)$ for $x > 0$.

The function $f(x) = (x^2 - 3)e^x$ has a local minimum at $x = \dots$

Find the minimum value of $f(x) = x + \frac{25}{x}$ for $x > 0$.

Find the $x$-coordinate of the point of inflection of $f(x) = 2x^3 - 3x^2 + 1$.

A point $P$ on the curve $y = f(x)$ where $f''(x)$ changes sign is called a:

What is the second derivative of $f(x) = \frac{1}{x}$ for $x \neq 0$?

For $f(x) = x^3 + 6x^2$, the graph is concave down on the interval:

The function $f$ is defined by $f(x) = x^4 + kx^3 + 6x^2$, where $k$ is a real constant. Given that the graph of $f$ is concave up for all $x \in \mathbb{R}$, find the possible values of $k$.

Consider the function $f(x) = x^2 \ln(x)$ for $x > 0$. The graph of $f$ has a point of inflection at $x = a$. Find the value of $a$.

The curve $y = 2x^3 + 3x^2 - 12x + 5$ is concave up when:

Find the $x$-coordinate of the point of inflection for $f(x) = x^3 \ln x$ for $x > 0$.

The point $(2, 4)$ is a point of inflection for $f(x) = x^3 + ax^2 + bx + 8$. Find the value of $a$.

For $f(x) = (x^2+1)e^x$, find the $x$-coordinates of the points of inflection.

What is the gradient of the tangent to $y = x^4$ at the point $(1, 1)$?

Find the gradient of the curve $y = \cos(2x)$ at $x = 0$.

The displacement of a particle is $s(t) = \frac{1}{t}$ for $t > 0$. Find the velocity at $t = 1$.

Find the gradient of the normal to the curve $y = (x + 2)^2$ at $x = -1$.

Find the equation of the normal to the curve $y = 2x^3$ at $x = 1$.

Find the value of $x$ for which the curve $y = \frac{x}{x^2+1}$ has a local maximum.

A particle moves such that $v(t) = t^3 - 3t^2 + 2t$. Find the acceleration when $t = 2$.

A particle moves along the $x$-axis with position $s(t) = t^3 - 6t^2 + 9t + 5$. Find the total distance traveled by the particle from $t = 0$ to $t = 4$.

For a gas obeying Boyle's Law, $PV = C$. If the volume is increasing at a rate of $10$ $\text{cm}^3\text{s}^{-1}$, find the rate of change of pressure when $P = 100$ kPa and $V = 20$ $\text{cm}^3$.

What is the maximum area of a right-angled triangle with a hypotenuse of length $10$ cm?

Find $\int \frac{1}{x^8} \, dx$.

Evaluate $\int (5x^4 - 3x^2) \, dx$.

What is $\int x^{4/3} \, dx$?

Find $\int (2x + 8) \, dx$.

Given that $f''(x) = 4 \cos(2x)$, and the graph of $y = f(x)$ has a gradient of $2$ at the point $(\frac{\pi}{4}, 1)$, find an expression for $f(x)$.

Find $\int (2x + e^x - \frac{1}{x}) dx$.

Calculate $\int \frac{x^2 - 9}{x+3} dx$.

A function $f$ is defined such that $f'(x) = (2x - 4)e^{x^2 - 4x}$. It is given that the minimum value of $f$ is $e^{-4} + 3$. Find the value of $f(0)$.

Let $f'(x) = \frac{\cos 2x}{\cos x + \sin x}$ for the domain $0 < x < \frac{\pi}{2}$. Given that the graph of $y = f(x)$ passes through the point $(\frac{\pi}{4}, \sqrt{2})$, find the value of $f(\frac{\pi}{2})$.

If $f'(x) = \cos(2x - \pi)$ and $f(\frac{\pi}{2}) = 1$, find $f(x)$.