Chain Rule: Explained with Examples, Master the Basics Today

Many functions come in layers: an outer function wrapped around an inner one, like (3x + 2)⁵ or sin(x²). The chain rule differentiates such nested functions. It says: differentiate the outer layer, keep the inner layer unchanged, then multiply by the derivative of the inner layer.

This matters because nesting is the most common way complicated functions are built. Exponential decay e^(−kt), oscillations sin(ωt), and nearly every formula in physics and engineering are compositions. Without the chain rule, differentiating them would force you back to first principles every time.

Key Concepts

• Inner and outer functions: in f(g(x)), g is the inner function and f is the outer function. Spot them before differentiating.
• Chain rule: d/dx [f(g(x))] = f'(g(x)) · g'(x). Outer derivative evaluated at the inner function, times the inner derivative.
• Multiple layers: for three nested functions, the rule extends: differentiate each layer outside-in and multiply all the pieces.
• Power of a function: d/dx [u(x)]ⁿ = n·[u(x)]ⁿ⁻¹ · u'(x) — the power rule plus the chain rule's extra factor.

Formulas

• Chain rule: if y = f(u) and u = g(x), then dy/dx = dy/du · du/dx.
• Function form: d/dx f(g(x)) = f'(g(x)) · g'(x).
• Power chain: d/dx (ax + b)ⁿ = n·a·(ax + b)ⁿ⁻¹.
• Common chains: d/dx e^(u) = e^(u)·u'; d/dx sin(u) = cos(u)·u'; d/dx ln(u) = u'/u.

Worked Example

Differentiate y = (3x + 2)⁵ and evaluate y'(0).

Identify the layers: the outer function is ( )⁵ and the inner function is u = 3x + 2. Differentiate the outer with respect to u: d/du [u⁵] = 5u⁴. Keep the inner as it is: 5(3x + 2)⁴. Now multiply by the inner derivative: du/dx = 3. The chain rule gives y' = 5(3x + 2)⁴ · 3 = 15(3x + 2)⁴. At x = 0: the inner value is 3(0) + 2 = 2, and 2⁴ = 16, so y'(0) = 15 · 16 = 240. Forgetting the final multiplication by 3 is the classic error — without it you would get 80, which is wrong.

Common Misconceptions

• Forgetting the inner derivative. The most common chain-rule mistake is stopping at f'(g(x)) and skipping the × g'(x) step.
• Evaluating the outer derivative at x. The outer derivative must be evaluated at the inner function g(x), not at x itself.
• Using the chain rule for products. Multiplication needs the product rule; nesting needs the chain rule. (3x + 2)⁵ is nesting, while x²(3x + 2) is a product.

Real-World Applications

• Exponential decay: differentiating e^(−kt) for radioactive decay and cooling uses the chain rule.
• Oscillations: velocity from position sin(ωt) needs the chain rule's extra factor ω.
• Related rates: a balloon's volume changing through its radius is a chain of dependencies.
• Neural networks: backpropagation is the chain rule applied across layers of a network.

In short, the chain rule handles functions inside functions: outer derivative at the inner function, times the inner derivative. Name the layers, work outside-in, and never skip that final multiplication — it is where most of the marks are lost.