Composition of Functions and the Chain Rule in Business Calculus

Composition of Functions

Definition and Notation

In calculus, a composite function is formed when the output of one function becomes the input of another. If g: A \rightarrow B and f: B \rightarrow C, then the composite function f \circ g: A \rightarrow C is defined by:

• Formula: $(f \circ g)(a) = f(g(a))$ for all $a \in A$

• Order matters: g is applied first, then f.

Example: If $f(x) = \sqrt{x}$ and $g(x) = x^2 + 3$, then:

• $(f \circ g)(x) = f(g(x)) = \sqrt{x^2 + 3}$

• $(g \circ f)(x) = g(f(x)) = (\sqrt{x})^2 + 3 = x + 3$

• Domains: $f \circ g$ has domain $\mathbb{R}$, $g \circ f$ has domain $[0, \infty)$

Note: In general, $f \circ g \neq g \circ f$.

Applications and Decomposition

Many complex functions can be viewed as compositions of simpler functions. This is useful for understanding their structure and for applying calculus techniques.

• Example: $F(x) = \frac{1}{\sqrt{2x + 3}}$ can be decomposed as:

  • $f(x) = 2x$

  • $g(x) = x + 3$

  • $h(x) = \sqrt{x}$

  • $i(x) = \frac{1}{x}$

  • $F(x) = (i \circ h \circ g \circ f)(x)$

The Chain Rule

Statement and Formula

The chain rule is a fundamental tool for differentiating composite functions. If $F(x) = f(g(x))$ and both $f$ and $g$ are differentiable, then:

• Chain Rule Formula: $F'(x) = f'(g(x)) \cdot g'(x)$

• In Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ where $y = f(u)$ and $u = g(x)$

Examples

• Example 1: $F(x) = (3x - 5)^{100}$

  • Let $f(x) = x^{100}$, $g(x) = 3x - 5$

  • $F'(x) = 100(3x - 5)^{99} \cdot 3 = 300(3x - 5)^{99}$

• Example 2: $F(x) = \frac{1}{\sqrt{x^2 + 2}}$

  • Let $f(x) = x^{-1/2}$, $g(x) = x^2 + 2$

  • $F'(x) = -\frac{1}{2}(x^2 + 2)^{-3/2} \cdot 2x = -\frac{x}{(x^2 + 2)^{3/2}}$

Chain Rule with Power Rule

When differentiating a composite function where the outer function is a power, combine the chain rule with the power rule:

• Formula: $\frac{d}{dx} [g(x)]^r = r [g(x)]^{r-1} \cdot g'(x)$

Chain Rule for Multiple Functions

The chain rule can be extended to compositions of more than two functions. If $F(x) = f(g(h(x)))$, then:

• $F'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

Evaluating Derivatives at a Point

When evaluating the derivative of a composite function at a specific value $x = a$:

• If $F(x) = f(g(x))$, then $F'(a) = f'(g(a)) \cdot g'(a)$

• Example using a table:

x	f(x)	g(x)	f'(x)	g'(x)
1	2	4	5	9
2	3	7	6	4
3	7	9	2	7

• $(f \circ g)'(1) = f'(g(1)) \cdot g'(1) = f'(4) \cdot 9 = 5 \cdot 9 = 45$

• $(g \circ f)'(1) = g'(f(1)) \cdot f'(1) = g'(2) \cdot 4 = 4 \cdot 4 = 16$

Related Rates (Preview)

The chain rule is essential in solving related rates problems, where two or more quantities change with respect to time.

• Example: A balloon is being inflated at a constant rate of $1 \text{ ft}^3/\text{sec}$. How fast is the radius increasing when it equals 1 foot?

• Volume of a sphere: $V = \frac{4}{3}\pi r^3$

• Given: $\frac{dV}{dt} = 1$

• By the chain rule: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$

• When $r = 1$, $1 = 4\pi (1)^2 \frac{dr}{dt} \implies \frac{dr}{dt} = \frac{1}{4\pi}$ ft/sec

Summary Table: Chain Rule Applications

Function	Derivative using Chain Rule
$(3x-5)^{100}$	$100(3x-5)^{99} \cdot 3$
$\frac{1}{\sqrt{x^2+2}}$	$-\frac{x}{(x^2+2)^{3/2}}$
$\sqrt{x^2+2}$	$\frac{x}{\sqrt{x^2+2}}$
$(x^2+x+1)^{10}$	$10(x^2+x+1)^9 \cdot (2x+1)$

Key Points

• Composite functions are formed by applying one function to the result of another.

• The chain rule allows differentiation of composite functions.

• Order of composition matters: $f \circ g \neq g \circ f$ in general.

• The chain rule can be extended to more than two functions.

• Related rates problems use the chain rule to relate rates of change of different quantities.