Lecture #12: The Chain Rule

Composition of Functions

Recall when we talked about composition of functions: If \(f\) and \(g\) are functions such that \(g : A \to B\) and \(f : B \to C\), we define the composition of \(f\) and \(g\) to be the new function \(h : A \to C\) give by \[ h(t) = f(g(t)). \]

Example: Composition of Functions

Example: For the following expressions, which is the “outer function”, \(f\) and which is the “inner function”, \(g\)?

1. \((5x^4 + 3x^7)^9\).
2. \(\sqrt{3x^2 + 2x + 1}\).
3. \(\dfrac{2}{x^3 + x^6}\).
4. \(\sin(x^2)\).

The Chain Rule

If \(f\) and \(g\) are differentiable functions, then: \[ \frac{d}{dx} (f \circ g)(x) = \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x). \]

Proof of the Chain Rule

\[ \begin{aligned} \frac{d}{dx} &f(g(x)) = \lim\limits_{h \to 0} \left[ \frac{f(g(x + h)) - f(g(x))}{h} \right] \\ &= \lim\limits_{h \to 0} \left[ \left( \frac{f(g(x + h)) - f(g(x))}{h} \right) \left( \frac{g(x + h) - g(x)}{g(x + h) - g(x)} \right) \right] \\ &= \lim\limits_{h \to 0} \left[ \left( \frac{f(g(x + h)) - f(g(x))}{g(x + h) - g(x)} \right) \left( \frac{g(x + h) - g(x)}{h} \right) \right] \\ &= \lim\limits_{h \to 0} \left[ \frac{f(g(x + h)) - f(g(x))}{g(x + h) - g(x)} \right] \times \lim\limits_{h \to 0} \left[ \frac{g(x + h) - g(x)}{h} \right] \\ &= f'(g(x)) \times g'(x) \end{aligned} \]

Example: The Chain Rule

Example: Find the derivative of the following functions:

1. \(f(x) = (5x^4 + 3x^7)^9\).
2. \(g(x) = \sqrt{3x^2 + 2x + 1}\).
3. \(h(x) = \left(x^4 + \dfrac{3}{x} \right)^3 + \sqrt{x}\).
4. \(q(x) = \dfrac{2}{\sqrt{x^2 + 3x}}\).
5. \(\ell(x) = \sqrt[3]{(x^2 + 4x)(x^5 + 7x^3)}\).

Example: The Chain Rule

Example: If \(f(x) = (4x^3 + 5x^2)^8\), find \(f'(x)\) and \(f''(x)\).