Why We Need Rules
Computing every derivative from the limit definition is correct but brutally slow. Calculus provides a toolkit of rules — each proved once from the definition, then usable forever. Learn these four rules fluently and you can differentiate almost anything instantly.
Power Rule — for xⁿ terms · Sum Rule — for adding functions · Product Rule — for multiplying functions · Quotient Rule — for dividing functions. The Chain Rule (next article) handles compositions.
The most-used rule in calculus. Bring the exponent down as a coefficient, then reduce the exponent by 1. Works for any real exponent n — positive, negative, fractional.
Differentiate term by term. The derivative of a sum is the sum of derivatives. This is what makes polynomials so easy to differentiate — just apply the Power Rule to each term.
For the product of two functions: first times derivative of second, plus second times derivative of first. A common mnemonic: "first d-second + second d-first."
⚠️ Common mistake: (fg)' ≠ f'g'. You cannot just multiply the derivatives.
For a ratio of two functions. Mnemonic: "lo d-hi minus hi d-lo, over lo squared" (lo = denominator g, hi = numerator f).
Why Rules, Not the Limit Definition Every Time?
The limit definition of the derivative is the rigorous foundation — but computing d/dx[x¹⁰⁰] from first principles would require expanding (x+h)¹⁰⁰ using the binomial theorem, a page-long calculation. The Power Rule gives the answer in one step: 100x⁹⁹. Each differentiation rule was proved once from the limit definition, then enshrined as a shortcut valid for all qualifying functions. Learning the rules — and their conditions — is what makes calculus computationally practical.
Proving the Product Rule
The Product Rule is not obvious, and understanding its proof deepens your ability to use it correctly. Starting from the limit definition: (fg)'(x) = lim(h→0) [f(x+h)g(x+h) − f(x)g(x)] / h. Add and subtract f(x+h)g(x) in the numerator: = lim [f(x+h)g(x+h) − f(x+h)g(x) + f(x+h)g(x) − f(x)g(x)] / h = lim f(x+h)·[g(x+h)−g(x)]/h + lim g(x)·[f(x+h)−f(x)]/h. As h→0: f(x+h) → f(x) (continuity), [g(x+h)−g(x)]/h → g'(x), [f(x+h)−f(x)]/h → f'(x). Result: f(x)g'(x) + g(x)f'(x) = f'g + fg'. This is why the rule takes the form it does.
Extending the Product Rule to Three or More Functions
For a product of three functions: (fgh)' = f'gh + fg'h + fgh'. The pattern: differentiate each factor once, leaving all others unchanged, and sum the results. For n functions: (f₁f₂···fₙ)' = Σᵢ f₁···fᵢ'···fₙ (differentiate one factor at a time, sum over all choices). This generalisation is occasionally needed for complex expressions and is easily derived by applying the two-function Product Rule iteratively.
Combining Multiple Rules — Complex Examples
The Constant Multiple and Sum Rules — Why They Work
d/dx[cf(x)] = c·f'(x) follows from the limit definition: c·f(x) changes at rate c times the rate of f(x). d/dx[f+g] = f'+g' follows because limits distribute over sums. These two rules together mean differentiation is a linear operator — a concept that becomes central in differential equations and functional analysis. In matrix form for polynomials, differentiation can be represented as a matrix acting on the vector of coefficients: d/dx[a₀ + a₁x + a₂x² + a₃x³] = a₁ + 2a₂x + 3a₃x², which corresponds to multiplying the coefficient vector by the "differentiation matrix."
When Rules Break Down
The Product, Quotient, and Power Rules all require the functions to be differentiable at the point in question. The Power Rule d/dx[xⁿ] requires n to be a constant — it does not apply to xˣ (where the exponent also depends on x). For xˣ, use logarithmic differentiation: ln y = x ln x → (1/y)(dy/dx) = ln x + 1 → dy/dx = xˣ(ln x + 1). The Quotient Rule requires g(x) ≠ 0 at the point of differentiation. Always check these conditions before applying rules mechanically.
- Stewart, J. (2015). Calculus, §3.1–3.4. Cengage.
- Apostol, T. (1967). Calculus, Vol. 1, Ch. 4. Wiley.
- Larson, R. & Edwards, B. (2013). Calculus, §2.2–2.5. Cengage.
To see all these rules tested in exam conditions, see the AP Calculus Review guide. Work through practice problems to build fluency, and consult the Common Mistakes guide to avoid the errors that cost marks.