Calculus

Derivatives, integrals, limits and series

Calculus is the toolkit for understanding change, accumulation, and how quantities behave over time, with formulas for derivatives, integrals, limits, and series all in one place. Whether you’re a student revising for exams, solving homework, or using maths in science, engineering, economics, or data work, these rules matter because they let you model motion, growth, rates, and area with precision.

Power Rule (Derivative)

\frac{d}{dx}x^n = n x^{n-1}

Derivative of a power function.

Chain Rule

\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)

Derivative of a composite function.

Product Rule

(fg)' = f'g + fg'

Derivative of a product of two functions.

Quotient Rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Derivative of a quotient of two functions.

Derivative of Sine

\frac{d}{dx}\sin x = \cos x

Derivative of the sine function.

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

Derivative of the cosine function.

Derivative of e^x

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

Derivative of ln x

\frac{d}{dx}\ln x = \frac{1}{x}

Derivative of the natural logarithm.

Power Rule (Integral)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

Integration by Parts

\int u\,dv = uv - \int v\,du

Integration technique from the product rule.

Fundamental Theorem of Calculus

\int_a^b f(x)\,dx = F(b) - F(a)

Connects differentiation and integration.

Derivative (Limit Definition)

f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}

Formal definition of the derivative.

Taylor Series

f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n

Expands a function as an infinite polynomial around a.

Maclaurin Series for e^x

e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}

Taylor series of the exponential at 0.

L'Hôpital's Rule

\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}

Evaluates indeterminate limits 0/0 or ∞/∞.

Arc Length (Integral)

L = \int_a^b \sqrt{1 + (f'(x))^2}\,dx

Length of a curve y = f(x) from a to b.

Surface of Revolution

S = 2\pi\int_a^b f(x)\sqrt{1+(f'(x))^2}\,dx

Surface area generated by revolving a curve about the x-axis.

Constant Rule (Derivative)

\frac{d}{dx}(c) = 0

The derivative of any constant is zero.

Sum Rule (Derivative)

(f+g)' = f' + g'

The derivative of a sum is the sum of the derivatives.

Constant Multiple Rule

\frac{d}{dx}(c\,f) = c\,f'

Constants factor out of a derivative.

Second Derivative

f''(x) = \frac{d}{dx}\left(f'(x)\right)

The derivative of the first derivative; measures concavity.

Derivative of Tangent

\frac{d}{dx}\tan x = \sec^2 x

Derivative of the tangent function.

Definite Integral (Area)

\int_a^b f(x)\,dx = \text{signed area under } f

The net area between a curve and the x-axis from a to b.

Average Value of a Function

f_{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx

The mean value of a continuous function over an interval.

Power Rule

\frac{d}{dx}x^n = nx^{n-1}

Derivative of a power function.

Product Rule

(uv)' = u'v + uv'

Derivative of a product of two functions.

Quotient Rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of a quotient of two functions.

Chain Rule

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

Derivative of a composite function.

Derivative of Sine

\frac{d}{dx}\sin x = \cos x

The derivative of sine is cosine.

Derivative of Cosine

\frac{d}{dx}\cos x = -\sin x

The derivative of cosine is negative sine.

Derivative of Exponential

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

Derivative of Natural Log

\frac{d}{dx}\ln x = \frac{1}{x}

The derivative of the natural logarithm.

Power Rule for Integration

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

Integral of Exponential

\int e^x\,dx = e^x + C

Antiderivative of the exponential function.

Integral of 1/x

\int \frac{1}{x}\,dx = \ln|x| + C

Antiderivative of the reciprocal function.

Mean Value Theorem

f'(c) = \frac{f(b)-f(a)}{b-a}

Guarantees a point where the tangent equals the average slope.

Second Derivative Test

f''(x) > 0 \Rightarrow \text{min}

Concavity determines local minima and maxima.

Average Value of a Function

\bar{f} = \frac{1}{b-a}\int_a^b f(x)\,dx

Mean value of a continuous function over an interval.

Volume by Disks

V = \pi\int_a^b [f(x)]^2\,dx

Volume of a solid of revolution about the x-axis.

Arc Length

L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx

Length of a curve between two points.

Partial Derivative

\frac{\partial f}{\partial x}

Rate of change with respect to one variable, holding others fixed.

Gradient Vector

\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)

Vector of partial derivatives pointing in the steepest ascent.

Fourier Transform

\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

Decomposes a function or signal into the frequencies that make it up — converting between the time (or space) domain and the frequency domain.