Chain Rule

Differentiate the outer function, then multiply by the derivative of the inner function.

Example

Differentiate \(y=(x^2+1)^{10}\) with respect to \(x\).

Let \(u=x^2+1\), so \(y=u^{10}\).

\(\dfrac{\mathrm{d}u}{\mathrm{d}x}=2x\qquad\dfrac{\mathrm{d}y}{\mathrm{d}u}=10u^9\)

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)\({}=\dfrac{\mathrm{d}y}{\mathrm{d}u}\dfrac{\mathrm{d}u}{\mathrm{d}x}\)\({}=10u^9(2x)\)\({}=20x(x^2+1)^9\)

For the smaller power \(y=(x^2+1)^2\), expansion is another way to differentiate.

\(y\)\({}=x^4+2x^2+1\)

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=4x^3+4x\)

Why the rates multiply

If \(u\) changes \(3\) times as fast as \(x\), and \(y\) changes \(4\) times as fast as \(u\), then \(y\) changes \(3\times4=12\) times as fast as \(x\). The chain rule multiplies these two rates of change.