Calculus Formulas

L'Hôpital's Rule

Evaluate indeterminate limits using L'Hôpital's Rule. Learn when and how to apply the derivative-based limit technique with examples.

Integration by Parts Formula

Integration by parts converts a hard integral into a simpler one. Learn the LIATE rule, formula derivation, and worked examples with polynomials and trig.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus explained — both parts, with examples showing how differentiation and integration are inverse operations.

Implicit Differentiation Formula

Implicit differentiation explained with step-by-step examples. Find dy/dx when y cannot be isolated, using the chain rule on both sides.

Partial Derivatives Formula

Partial derivatives explained with notation, rules, and examples. Learn how to differentiate multivariable functions with respect to one variable.

Arc Length Formula

Reference for arc length L = integral of sqrt(1 + (dy/dx)^2) dx. Covers Cartesian and parametric forms with worked examples in engineering and physics.

Laplace Transform

Reference for the Laplace Transform L{f(t)} = integral of e^(-st)f(t)dt. Includes common pairs for exponential, sinusoidal, step, and polynomial functions.

Mean Value Theorem

Understand the Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a). Guarantees a point where instantaneous rate equals average rate.

Surface Area of Revolution Formula

Calculate the surface area of revolution using S = 2pi * integral f(x) sqrt(1+(f'(x))^2) dx. Covers rotation about x and y axes with worked examples.

Area Between Curves

Reference for the area between curves formula A = ∫[f(x) - g(x)]dx over [a,b]. Covers vertical vs horizontal slicing and finding intersection points.

Chain Rule

Reference for the chain rule d/dx[f(g(x))] = f'(g(x))*g'(x). Step-by-step examples with trig, exponential, and nested polynomial composite functions.

Derivative Rules

Complete reference of derivative rules: power rule, product rule, quotient rule, chain rule, and common derivatives. Includes worked examples for each rule.

Integration Rules

Complete integration formula reference covering the power rule, substitution, integration by parts, and common integrals. Worked examples for each rule.

Limit Properties and Rules

Reference for essential limit laws and properties in calculus. Covers sum, product, quotient, and power rules, L'Hopital's Rule, and the squeeze theorem.

Product and Quotient Rules

Reference for the product rule (fg)' = fg'+gf' and quotient rule (f/g)' = (gf'-fg')/g^2. Side-by-side worked examples and tips for choosing the right rule.

Taylor and Maclaurin Series

Reference for Taylor and Maclaurin series expansions. Covers common series for e^x, sin(x), cos(x), ln(1+x), and the geometric series with convergence radii.

Volume of Revolution (Disk and Shell Methods)

Formulas for calculating volumes of solids of revolution using the disk method, washer method, and cylindrical shell method.