Derivatives: The Slope of a Function
The derivative measures how fast a function changes at any point — it's the instantaneous rate of change and the slope of the tangent line.
Key Concepts
Definition of the Derivative
f'(x) = lim[h→0] (f(x+h) − f(x)) / h. This is the limit of the difference quotient — the slope of a secant line as it approaches a tangent.
Power Rule
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Examples: (x³)' = 3x², (x⁵)' = 5x⁴, (x)' = 1, (constant)' = 0. The most-used rule in calculus.
Sum, Product & Chain Rules
Sum: (f+g)' = f'+g'. Product: (fg)' = f'g + fg'. Chain: if h(x)=f(g(x)), then h'(x) = f'(g(x))·g'(x). Used for composite functions.
Applications
Derivative = 0 at max/min (critical points). Positive derivative = increasing. Negative = decreasing. Second derivative f'' tells concavity and inflection points.
Live Python Practice
Interactive Lab
Check Your Understanding
The derivative of x⁴ is:
f'(x) = 0 at a point means:
The definition of f'(x) uses: