Understanding the Chain Rule in Differentiation

The chain rule is a fundamental theorem in calculus that allows us to differentiate composite functions. A composite function is a function that is made up of two or more functions, where the output of one function becomes the input of another.

The Chain Rule Statement

The chain rule states that if we have two functions $f(x)$ and $g(x)$, and a composite function $h(x) = f(g(x))$, then the derivative of $h(x)$ with respect to $x$ is given by:

$$ \frac{dh}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx} $$

In other words, to find the derivative of the composite function $h(x)$, we first take the derivative of the outer function $f(g(x))$ with respect to the inner function $g(x)$, and then multiply it by the derivative of the inner function $g(x)$ with respect to $x$.

Important Points and Differences

Aspect	Description
Function Composition	The process of combining two functions where the output of one becomes the input of the other.
Outer Function	In a composite function $f(g(x))$, $f$ is the outer function.
Inner Function	In a composite function $f(g(x))$, $g$ is the inner function.
Derivative	The rate at which a function is changing at any given point.
Composite Function	A function formed by combining two or more functions.

Formulas

The chain rule can be expressed in Leibniz notation as:

$$ \frac{d}{dx}[f(g(x))] = \frac{df}{dg} \cdot \frac{dg}{dx} $$

Or in prime notation as:

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

Examples

Example 1: Basic Application

Let's say we have a function $h(x) = (3x + 2)^2$. To differentiate $h(x)$ using the chain rule, we identify the inner function $g(x) = 3x + 2$ and the outer function $f(u) = u^2$ where $u = g(x)$.

1. Differentiate the outer function with respect to $u$:

$$ \frac{df}{du} = 2u $$

1. Differentiate the inner function with respect to $x$:

$$ \frac{dg}{dx} = 3 $$

1. Apply the chain rule:

$$ \frac{dh}{dx} = \frac{df}{du} \cdot \frac{dg}{dx} = 2u \cdot 3 = 2(3x + 2) \cdot 3 = 6(3x + 2) $$

Example 2: Trigonometric Function

Consider $h(x) = \sin(x^2)$. Here, the inner function is $g(x) = x^2$ and the outer function is $f(u) = \sin(u)$.

1. Differentiate the outer function with respect to $u$:

$$ \frac{df}{du} = \cos(u) $$

1. Differentiate the inner function with respect to $x$:

$$ \frac{dg}{dx} = 2x $$

1. Apply the chain rule:

$$ \frac{dh}{dx} = \frac{df}{du} \cdot \frac{dg}{dx} = \cos(u) \cdot 2x = \cos(x^2) \cdot 2x $$

Example 3: Exponential Function

Let's differentiate $h(x) = e^{(5x + 1)}$. The inner function is $g(x) = 5x + 1$ and the outer function is $f(u) = e^u$.

1. Differentiate the outer function with respect to $u$:

$$ \frac{df}{du} = e^u $$

1. Differentiate the inner function with respect to $x$:

$$ \frac{dg}{dx} = 5 $$

1. Apply the chain rule:

$$ \frac{dh}{dx} = \frac{df}{du} \cdot \frac{dg}{dx} = e^u \cdot 5 = 5e^{(5x + 1)} $$

In conclusion, the chain rule is a powerful tool for differentiating composite functions. By breaking down the function into its inner and outer components, we can systematically find the derivative by applying the rule. Understanding and practicing the chain rule is essential for success in calculus and related fields.