Formula Reference

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Power Rule

d/dx(x^n) = nx^(n-1)

Derivative of a power function

Chain Rule

d/dx(f(g(x))) = f'(g(x))·g'(x)

Derivative of composite functions

Fundamental Theorem

∫[a to b] f'(x)dx = f(b) - f(a)

Connects derivatives and integrals

All Formulas

Power Rule

derivatives
d/dx(x^n) = nx^(n-1)

Derivative of a power function

Example:

d/dx(x³) = 3x²

Product Rule

derivatives
d/dx(uv) = u'v + uv'

Derivative of a product of functions

Example:

d/dx(x²sin(x)) = 2x·sin(x) + x²·cos(x)

Chain Rule

derivatives
d/dx(f(g(x))) = f'(g(x))·g'(x)

Derivative of composite functions

Example:

d/dx(sin(x²)) = cos(x²)·2x

Integration by Parts

integrals
∫u dv = uv - ∫v du

Integration technique for products

Example:

∫x·e^x dx = xe^x - e^x + C

Fundamental Theorem

integrals
∫[a to b] f'(x)dx = f(b) - f(a)

Connects derivatives and integrals

Example:

∫[0 to 1] 2x dx = x²|₀¹ = 1

L'Hôpital's Rule

limits
lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)

For indeterminate forms 0/0 or ∞/∞

Example:

lim[x→0] sin(x)/x = lim[x→0] cos(x)/1 = 1

Geometric Series

series
∑[n=0 to ∞] ar^n = a/(1-r) for |r| < 1

Sum of infinite geometric series

Example:

∑[n=0 to ∞] (1/2)^n = 1/(1-1/2) = 2

Pythagorean Identity

trigonometry
sin²(x) + cos²(x) = 1

Fundamental trigonometric identity

Example:

sin²(π/4) + cos²(π/4) = (√2/2)² + (√2/2)² = 1