Derivative Rules
Complete reference of derivative rules: power rule, product rule, quotient rule, chain rule, and common derivatives.
Includes worked examples for each rule.
Basic Derivative Rules
d/dx [c] = 0 (constant rule)
d/dx [x^n] = n·x^(n-1) (power rule)
d/dx [c·f(x)] = c·f'(x) (constant multiple)
d/dx [f(x) + g(x)] = f'(x) + g'(x) (sum rule)
d/dx [x^n] = n·x^(n-1) (power rule)
d/dx [c·f(x)] = c·f'(x) (constant multiple)
d/dx [f(x) + g(x)] = f'(x) + g'(x) (sum rule)
Product and Quotient Rules
d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x) (product rule)
d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]² (quotient rule)
d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]² (quotient rule)
Chain Rule
d/dx [f(g(x))] = f'(g(x)) · g'(x)
The chain rule is used when differentiating a function of a function (composite functions).
Common Derivatives
| Function f(x) | Derivative f'(x) |
|---|---|
| x^n | n·x^(n-1) |
| e^x | e^x |
| a^x | a^x · ln(a) |
| ln(x) | 1/x |
| log_a(x) | 1/(x·ln(a)) |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| cot(x) | -csc²(x) |
| sec(x) | sec(x)·tan(x) |
| csc(x) | -csc(x)·cot(x) |
| arcsin(x) | 1/√(1-x²) |
| arccos(x) | -1/√(1-x²) |
| arctan(x) | 1/(1+x²) |
Example 1 — Power Rule
Find d/dx [3x⁵]
Apply: d/dx [ax^n] = a·n·x^(n-1)
= 3 · 5 · x^(5-1) = 15x⁴
Example 2 — Product Rule
Find d/dx [x² · sin(x)]
Let f(x) = x², g(x) = sin(x)
f'(x) = 2x, g'(x) = cos(x)
= 2x·sin(x) + x²·cos(x) = 2x·sin(x) + x²·cos(x)
Example 3 — Chain Rule
Find d/dx [sin(3x²)]
Outer function: sin(u), inner function: u = 3x²
= cos(3x²) · d/dx[3x²]
= cos(3x²) · 6x = 6x·cos(3x²)
Key Notes
- The chain rule is the most-used rule in practice — virtually every real-world function is a composition; the key skill is identifying the outer and inner functions before differentiating
- The quotient rule can always be replaced by the product rule: treat f/g as f × g⁻¹ and apply the chain rule to g⁻¹ — this sometimes produces fewer terms
- Implicit differentiation applies the chain rule to equations where y is not isolated: differentiate both sides with respect to x and multiply any dy/dx term explicitly — used for curves like x² + y² = 25
- Higher-order derivatives (f''(x), f'''(x)) represent acceleration, jerk, and beyond; the second derivative determines concavity — f'' > 0 means the curve is concave up (cup-shaped)