The Chain Rule

Subtopic: Differentiation

The chain rule differentiates compositions: if y depends on u and u depends on x, how does y depend on x? Multiply the rates. This simple principle handles nested functions, implicit differentiation, and related rates. It's essential whenever functions are built from other functions.

Introduction

You're driving. Your speed depends on the gear, and the gear depends on time. How does speed depend on time? Combine the effects: (change in speed per gear) ⋅ (change in gear per time).

The chain rule makes this precise: the derivative of a composition is the product of the derivatives.

The Formula

If y=ƒ(g(x)), then:

d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)=ƒ′*(g(x))⋅g(x)′

In words: differentiate the outer function (leaving the inner alone), then multiply by the derivative of the inner.

Leibniz notation makes it memorable: the d(u) "cancel" (though this is just mnemonic, not rigorous).

Why It Works

Small changes propagate through compositions. If x changes by Δ(x):

u=g(x) changes by approximately g′*(x)*Δ(x).

y=ƒ(u) changes by approximately ƒ′*(u)×(change in *u)=ƒ′*(g(x))g′*(x)*Δ(x).

The rates multiply.

Worked Example 1

Differentiate y=(3*x+1)5.

Let u=3*x+1, so y=u5.

d(y)/d(u)=5*u4, and d(u)/d(x)=3.

d(y)/d(x)=5*u4⋅3=15*(3*x+1)4

Multiple Compositions

For y=ƒ(g(h(x))), apply chain rule multiple times:

d(y)/d(x)=ƒ′*(g(h(x)))⋅g′*(h(x))⋅h′*(x)

Peel off layers one at a time, multiplying derivatives.

Common Patterns

These follow directly from the chain rule:

• d(ƒ(x)n)/d(x)=n*ƒ(x)(n-1)ƒ′*(x)

• d(eƒ(x))/d(x)=eƒ(x)ƒ′*(x)

• d(ln(ƒ(x)))/d(x)=(ƒ′*(x))/ƒ(x)

• d(sin(ƒ(x)))/d(x)=cos(ƒ(x))ƒ′*(x)

Implicit Differentiation

The chain rule powers implicit differentiation. If y is a function of x, then:

d(y2)/d(x)=2*yd(y)/d(x)

The d(y)/d(x) factor appears because we're differentiating a function of y with respect to x.

Related Rates

In related rates problems, quantities change with time. The chain rule connects their rates:

d(V)/d(t)=d(V)/d(r)d(r)/d(t)

If you know how volume depends on radius and how radius changes with time, you can find how volume changes with time.

Applications

In neural networks, back propagation applies the chain rule to compute gradients through many layers.

In physics, the chain rule relates velocities and accelerations in different coordinate systems.

In economics, it computes how changes in input prices affect output costs through intermediate goods.

Summary

The chain rule differentiates compositions: (ƒ∘g)′*(x)=ƒ′*(g(x))g′*(x). In Leibniz notation: d(y)/d(x)=d(y)/d(u)d(u)/d(x). Apply it from outside in: differentiate the outer function, then multiply by the derivative of the inner function. For multiple compositions, apply the rule repeatedly. The chain rule underlies implicit differentiation, related rates, and backpropagation.