Calculus

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.