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Derivative

Undergraduate

Definition

The instantaneous rate of change of a function, or slope of the tangent line.

f'(x) = lim_h → 0 (f(x+h) - f(x))/(h)

Definition of derivative (limit)

(d)/(dx)(xⁿ) = nx^n-1

Power rule

(fg)' = f'g + fg'

Product rule

((f)/(g))' = (f'g - fg')/(g²)

Quotient rule

Example 1

Find the derivative of f(x) = x³.

f'(x) = 3x²

Example 2

Find the derivative of f(x) = x²·sin(x).

f'(x) = 2xsin(x) + x²cos(x)

Discovered by: Isaac Newton, Gottfried Leibniz (Late 17th century)

Newton and Leibniz independently invented calculus.

Velocity, acceleration

Marginal cost, marginal revenue

Finding maxima and minima