4.3: The Chain Rule

In the last two sections we learned rules to symbolically differentiate some functions. To summarize, we have established some elementary formulas and some arithmetic rules.

Elementary Formulas:

If \(f(x)=a\text{,}\) then \(f'(x)=0\text{.}\)

If \(f(x)=ax\text{,}\) then \(f'(x)=a\text{.}\)

If \(f(x)=a*x^n\text{,}\) then \(f(x)=a*n*x^{n-1}\text{,}\) for any nonzero number n.

If \(f(x)=e^x\text{,}\) then \(f'(x)=e^x\text{.}\)

If \(f(x)=a^x\text{,}\) then \(f'(x)=a^x \ln(a)\text{.}\)

If \(f(x)=\ln(x)\text{,}\) then \(f'(x)=1/x\)

Arithmetic derivative rules:

Scalar multiple rule: The derivative of \(c*f(x)\) is \(c*f'(x)\text{.}\)

Sum and difference rule: The derivative of \(f(x)\pm g(x)\) is \(f'(x)\pm g'(x)\text{.}\)

Product Rule: The derivative of \(f(x)g(x)\) is \(f' (x) g(x)+f(x)g'(x)\text{.}\)

Quotient Rule: The derivative of \(f(x)/g(x)\) is \(\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2} \text{.}\)

The other way we traditionally build functions from simpler functions is by use of composition. We want to be able to take derivatives of functions like \((2x+3)^{52}\text{,}\) \(\sqrt{(x^2 )+5x+7}\text{,}\) and \(1.06^{.2x}\text{.}\)

Claim \(4.3.1\). Chain Rule.

The derivative of \(f(g(x))\) is \(f'(g(x))g'(x)\) In other words,

\[ \left[f(g(x))\right]'=f'(g(x))*g'(x)\text{.} \nonumber \]

Example 4.3.2: Simple Chain Rule.

Find the derivative of the following functions:

1. \(\displaystyle f(p)=(p^3+2 p+5)^7.\)
2. \(\displaystyle g(q)=\sqrt{q^2+6}.\)
3. \(\displaystyle h(x)=e^{2 x+5}.\)

Solution

1. We could do this problem by expanding it to a polynomial and using rules from the previous section, but that is much too hard. We can write \(f(p)\) as \(g(h(p))\) where \(h(p)=p^3+2 p+5\) and \(g(p)=p^7\text{.}\) We use the rules from the previous section to compute \(h'(p)=3 p^2+2\) and \(g'(p)=7p^6\text{.}\) Composing we get \(g'(h(p))=7(p^3+2 p+5)^6\text{.}\) Thus

   \[ f'(p)=g'(h(p))h'(p)=7(p^3+2 p+5)^6(3 p^2+2 ). \nonumber \]