Practice Maths

Product and Quotient Rules for All Function Types

Key Terms

Product rule
d/dx[u · v] = u′v + uv′.
Quotient rule
d/dx[u/v] = (u′v − uv′) / v².
When to use product rule
When f(x) = g(x) · h(x); identify the two factors u and v.
When to use quotient rule
When f(x) = g(x) / h(x); identify numerator u and denominator v.
Quotient rule mnemonic
“Low d-high minus high d-low, square the bottom and away we go.”
Combined with chain rule
Each factor u and v may itself require the chain rule when differentiating; apply rules in order.
Product Rule: d/dx[u · v] = u′v + uv′
If y = f(x)·g(x), then dy/dx = f′(x)g(x) + f(x)g′(x)

Quotient Rule: d/dx[u/v] = (u′v − uv′) / v²
If y = f(x)/g(x), then dy/dx = [f′(x)g(x) − f(x)g′(x)] / [g(x)]²

Common derivatives needed:
d/dx[xn] = nxn−1
d/dx[ex] = ex
d/dx[ln x] = 1/x
d/dx[sin x] = cos x
d/dx[cos x] = −sin x
Worked Example 1 (Product): Differentiate y = x² sin x.

Let u = x², v = sin x ⇒ u′ = 2x, v′ = cos x
dy/dx = u′v + uv′ = 2x sin x + x² cos x
Worked Example 2 (Quotient): Differentiate y = sin x / x.

Let u = sin x, v = x ⇒ u′ = cos x, v′ = 1
dy/dx = (cos x · x − sin x · 1) / x² = (x cos x − sin x) / x²
Hot Tip: For the quotient rule, the order of the numerator matters: it’s (top′ × bottom − top × bottom′) — never the other way around. A useful mnemonic is “lo d-hi minus hi d-lo over lo-squared”: (v u′ − u v′) / v² when written as u/v.

Why We Need Product and Quotient Rules

When a function is formed by multiplying or dividing two simpler functions, the derivative is NOT simply the product or quotient of their derivatives. For example, d/dx[x² · sin x] ≠ 2x · cos x. The product rule tells us the correct procedure: differentiate each factor in turn while leaving the other unchanged, then add the results.

The product rule can be remembered as: “first’ times second, plus first times second’”. Written out: if y = u · v, then dy/dx = u′v + uv′. This comes directly from the limit definition of the derivative applied to a product.

The Product Rule in Practice

Step 1: Identify u and v (the two factors being multiplied).

Step 2: Find u′ and v′ separately, including the chain rule if needed.

Step 3: Apply u′v + uv′ and simplify.

For y = ex ln x: u = ex, v = ln x, u′ = ex, v′ = 1/x.
dy/dx = ex ln x + ex · (1/x) = ex(ln x + 1/x).

For y = x3 cos x: u = x3, v = cos x, u′ = 3x², v′ = −sin x.
dy/dx = 3x² cos x − x3 sin x = x²(3 cos x − x sin x).

The Quotient Rule in Practice

For y = f/g: dy/dx = (f′g − fg′) / g². The denominator is always g² (the original denominator squared). Note the minus sign in the numerator — order matters!

For y = x3/ex: u = x3, v = ex, u′ = 3x², v′ = ex.
dy/dx = (3x² · ex − x3 · ex) / (ex)² = ex(3x² − x3) / e2x = x²(3 − x) / ex.

For y = ln x / x2: u = ln x, v = x², u′ = 1/x, v′ = 2x.
dy/dx = ((1/x) · x² − ln x · 2x) / x4 = (x − 2x ln x) / x4 = (1 − 2 ln x) / x3.

Combining Rules: Product + Chain Rule

Many problems require both the product rule and the chain rule together. For y = x² e−x: u = x², v = e−x. Note v′ = −e−x (chain rule). Then dy/dx = 2x e−x + x² (−e−x) = e−x(2x − x²) = xe−x(2 − x).

Setting this equal to zero: x = 0 or x = 2 are the stationary points. The factor e−x is never zero, so it doesn’t contribute stationary points.

When to Use Which Rule

  • Product rule: when two functions are multiplied, e.g. x sin x, ex ln x, x² cos(2x).
  • Quotient rule: when one function divides another, e.g. sin x / x, ex / x², ln x / cos x.
  • Note: sometimes a quotient can be rewritten as a product. For example, ex/x2 = ex · x−2, allowing product rule use. Both methods give the same answer.
Labelling strategy: Always label your u, v, u′, v′ explicitly before substituting into the rule. This dramatically reduces sign errors and missed terms, especially when chain rule is also involved.

Mastery Practice

  1. Differentiate y = x² sin x using the product rule.
  2. Differentiate y = ex ln x using the product rule.
  3. Differentiate y = sin x / x using the quotient rule.
  4. Differentiate y = x3 / ex using the quotient rule.
  5. Differentiate y = x cos x and find the gradient at x = π.
  6. Differentiate y = ex sin x and find the x-values in [0, 2π] where the gradient is zero.
  7. Differentiate y = ln x / x2 and simplify.
  8. Differentiate y = x2 e−x and find the stationary points.
  9. Find the equation of the tangent to y = x ex at x = 0.
  10. A function is defined by f(x) = (x² + 1) / (x² − 1) for x ≠ ±1. Find f′(x) and determine where f′(x) = 0.
See Answers ➔