Differentiation Rules: Complete Guide
Differentiation rules are the toolkit that makes finding derivatives systematic and fast — no need to return to the limit definition for every function. Six core rules cover virtually all standard functions encountered in calculus.
Master these rules in order: constant, power, sum/difference, constant multiple, then product and quotient rules, and finally the chain rule for composites. Each rule builds on the previous ones.
Formula
d/dx[xⁿ] = n·xⁿ⁻¹ | d/dx[f·g] = f'g+fg' | d/dx[f(g)] = f'(g)·g'
Basic Rules: Constant, Power, Sum, and Constant Multiple
Constant Rule: d/dx[c] = 0 — constants have zero rate of change. d/dx[7] = 0, d/dx[π] = 0.
Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹ for any real exponent n. Examples: d/dx[x⁴] = 4x³; d/dx[x^(1/2)] = (1/2)x^(−1/2) = 1/(2√x); d/dx[x⁻³] = −3x⁻⁴.
Constant Multiple Rule: d/dx[c·f(x)] = c·f'(x). Pull constants outside: d/dx[5x³] = 5·3x² = 15x².
Sum/Difference Rule: d/dx[f ± g] = f' ± g'. Differentiate term by term: d/dx[x⁴ − 3x² + 7] = 4x³ − 6x.
| Function | Derivative | Rule Applied |
|---|---|---|
| c (constant) | 0 | Constant Rule |
| xⁿ | n·xⁿ⁻¹ | Power Rule |
| c·f(x) | c·f'(x) | Constant Multiple |
| f(x) + g(x) | f'(x) + g'(x) | Sum Rule |
| f(x) − g(x) | f'(x) − g'(x) | Difference Rule |
| eˣ | eˣ | Exponential |
| aˣ | aˣ · ln a | Exponential (base a) |
| ln x | 1/x | Logarithm |
| sin x | cos x | Trigonometric |
| cos x | −sin x | Trigonometric |
| tan x | sec²x | Trig (via Quotient Rule) |
Product Rule and Quotient Rule
Product Rule: d/dx[f·g] = f'g + fg'. "Derivative of first times second, plus first times derivative of second." Example: d/dx[x²·sin x] = 2x·sin x + x²·cos x.
Quotient Rule: d/dx[f/g] = (f'g − fg') / g². "Low d-high minus high d-low, over low squared." Example: d/dx[sin x / x] = (x·cos x − sin x) / x².
A helpful check: the quotient rule is derivable from the product rule. d/dx[f/g] = d/dx[f · g⁻¹] = f'·g⁻¹ + f·(−g⁻²·g') = (f'g − fg') / g². If you forget the quotient rule formula, derive it.
Common mistake: treating (f·g)' = f'·g'. Counterexample: d/dx[x·x] = d/dx[x²] = 2x, not 1·1 = 1.
Chain Rule: Differentiating Composite Functions
Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x). Differentiate the outer function (leaving inner alone), multiply by the derivative of the inner function.
Identifying the composition: ask "what is computed last?" That's the outer function. Everything inside is the inner function.
Examples: d/dx[sin(3x²)] = cos(3x²) · 6x. d/dx[e^(x³)] = e^(x³) · 3x². d/dx[(4x+1)⁷] = 7(4x+1)⁶ · 4 = 28(4x+1)⁶.
Triple composition: d/dx[sin(e^(x²))] = cos(e^(x²)) · e^(x²) · 2x. Work from outside in, one layer at a time.
| Composite Function | Derivative | Outer · Inner' |
|---|---|---|
| (3x+2)⁵ | 5(3x+2)⁴ · 3 = 15(3x+2)⁴ | Power of inner |
| sin(x²) | cos(x²) · 2x | Trig of poly |
| e^(5x) | 5e^(5x) | Exp of linear |
| ln(x³+1) | 3x²/(x³+1) | Log of poly |
| √(2x−1) | 1/√(2x−1) | Root of linear |
| cos(sin x) | −sin(sin x) · cos x | Trig of trig |
How to Choose the Right Rule
Look at the outermost structure of the expression: (1) Single power of x → Power Rule. (2) Sum of terms → differentiate each separately. (3) Two functions multiplied → Product Rule. (4) One function divided by another → Quotient Rule. (5) Function of a function → Chain Rule.
Complex expressions often require combinations. d/dx[x²·sin(3x)]: outer structure is a product (x² times sin(3x)) → Product Rule. But sin(3x) is itself a composite → Chain Rule needed for that factor.
Worked combination: d/dx[x²·sin(3x)] = 2x·sin(3x) + x²·[cos(3x)·3] = 2x·sin(3x) + 3x²·cos(3x). Product rule gives the structure; chain rule handles sin(3x).