Understanding the Chain Rule in Calculus: A Comprehensive Guide to Calculating Derivatives of Composite Functions

chain rule

f'(g(x))g'(x)

The chain rule is a fundamental rule in calculus that allows us to calculate the derivative of a composite function. It is used when we have a function within another function, i.e., a composition of functions.

To understand the chain rule, let’s consider two functions: f(x) and g(x). If we have a composite function h(x) = f(g(x)), the chain rule states that its derivative, denoted as h'(x), can be calculated as the product of the derivatives of the outer and inner functions.

In mathematical notation, the chain rule can be stated as follows:

h'(x) = f'(g(x)) * g'(x)

The chain rule is important because it allows us to find the derivative of complicated functions by breaking them down into simpler functions and applying the chain rule successively.

To apply the chain rule, follow these steps:

1. Identify the outermost function, which is typically denoted as f(x).
2. Identify the inner function, which is typically denoted as g(x).
3. Compute the derivative of the outer function f'(x).
4. Compute the derivative of the inner function g'(x).
5. Substitute the derivative values into the formula h'(x) = f'(g(x)) * g'(x).

Let’s illustrate the chain rule with an example:

Consider the function h(x) = (x^2 + 3x)^3. To find its derivative, we need to apply the chain rule.

1. Identify the outer function, f(x), as (x^3).
2. Identify the inner function, g(x), as (x^2 + 3x).
3. Compute the derivative of the outer function: f'(x) = 3x^2.
4. Compute the derivative of the inner function: g'(x) = 2x + 3.
5. Substitute the derivative values into the chain rule formula:
h'(x) = f'(g(x)) * g'(x)
= 3(x^2 + 3x)^2 * (2x + 3)

Thus, the derivative of h(x) = (x^2 + 3x)^3 is h'(x) = 3(x^2 + 3x)^2 * (2x + 3).

The chain rule is a powerful tool in calculus, enabling us to differentiate complex functions by breaking them down into simpler components.

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