The derivative of a function measures its instantaneous rate of change: the slope of the tangent line at a point. Formally, f′(x) = lim (h→0) [f(x + h) − f(x)] / h. In practice you rarely use the limit directly; you use a handful of rules.
Constant and power rules
The derivative of a constant is 0. For any real n, d/dx xⁿ = n·xⁿ⁻¹. Example: d/dx (3x² + 2x) = 6x + 2.
Product rule
(uv)′ = u′v + uv′. Example: for y = x²·sin x, y′ = 2x·sin x + x²·cos x.
Quotient rule
(u/v)′ = (u′v − uv′) / v². Example: for y = x / (x + 1), y′ = [(1)(x + 1) − x(1)] / (x + 1)² = 1 / (x + 1)².
Chain rule
For a function inside a function, differentiate the outside and multiply by the derivative of the inside: d/dx f(g(x)) = f′(g(x))·g′(x). Example: d/dx sin(x²) = cos(x²)·2x.
Derivatives worth memorizing
d/dx sin x = cos xandd/dx cos x = −sin xd/dx eˣ = eˣd/dx ln x = 1/xfor x > 0
How to check your work
Plug a nearby value into the original function and estimate the slope: for f(x) = x² at x = 3, [f(3.001) − f(3)] / 0.001 ≈ 6.001, which matches f′(3) = 6.