Differentiation Rules
Once you understand what a derivative means conceptually, differentiation rules let you calculate derivatives quickly without going back to the limit definition every time. These four rules handle the vast majority of functions you will encounter.
1 The Power Rule
For f(x) = xⁿ: f'(x) = nxⁿ⁻¹. Bring the exponent down as a coefficient, reduce the exponent by 1.
For sums and differences: differentiate term by term. d/dx[f(x) + g(x)] = f'(x) + g'(x). For constants multiplied by functions: d/dx[cf(x)] = c·f'(x).
2 The Product Rule
For f(x) = u(x)·v(x): f'(x) = u'v + uv'
"First times derivative of second, plus second times derivative of first." The derivative of a product is NOT the product of the derivatives.
d/dx[x² · x³] ≠ 2x · 3x². The correct answer uses the product rule (or just simplify first: x⁵ → 5x⁴). Always simplify first if possible.
3 The Quotient Rule
For f(x) = u(x)/v(x): f'(x) = (u'v − uv') / v²
Memory trick: "Low d-High minus High d-Low, all over Low squared." (Low = denominator, High = numerator, d = derivative)
4 The Chain Rule
For composite functions f(g(x)): d/dx[f(g(x))] = f'(g(x)) · g'(x)
"Derivative of the outside (keeping inside unchanged), times derivative of the inside."
10x(x²+3)⁴Any time you see a function raised to a power, a trig function of an expression, or a composition of functions, the chain rule applies. The signal: 'there's something inside something else.'
5 Combining the Rules
Real problems often require multiple rules together. The key is identifying the structure: what's the outermost operation? Apply that rule first, then handle the inner parts.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.