Derivative Chain Rule

The chain rule gives the derivative of a composition of functions.

Statement

If $y=f(g(x))$, then

$$\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$$

Another way this is written is

$$\frac{dy}{dx}=\frac{dy}{dg}\cdot \frac{dg}{dx}$$

Derivation

The derivation of the chain rule using the Limit Definition of the Derivative.

Beginning with the limit definition

$$\frac{d}{dx}f(g(x))=\underset{h\to 0}{\lim }\frac{f(g(x+h))-f(g(x))}{h}$$

Multiply and divide by $g(x+h)-g(x)$

$$\frac{d}{dx}f(g(x))=\underset{h\to 0}{\lim }\frac{f(g(x+h))-f(g(x))}{g(x+h)-g(x)}\cdot \frac{g(x+h)-g(x)}{h}$$

Now take the limits of each part separately

• The first term becomes $f^{\prime }(g(x))$
• The second term becomes $g^{\prime }(x)$

Thus,

$$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)$$