Master the Chain Rule: Derivatives of Composite Functions in Calculus

Chain Rule

The chain rule is a rule in calculus that enables us to find the derivative of a composite function

The chain rule is a rule in calculus that enables us to find the derivative of a composite function. A composite function is a function that is composed of two or more functions.

To understand the chain rule, let’s consider an example. Let’s say we have a function f(x) = g(h(x)), where g(x) and h(x) are both functions. In this case, g is the outer function, and h is the inner function.

To find the derivative of f(x), we need to find the derivative of the outer function, g, with respect to the inner function, h, and then multiply it with the derivative of the inner function, h, with respect to x (dx).

Mathematically, the chain rule can be expressed as follows:

d(f(g(x)))/dx = (df(g(x))/dg(x)) * (dg(x)/dx)

Let’s break down the chain rule with an example using specific functions:

Example: Find the derivative of f(x) = (2x + 1)^3 using the chain rule.

Step 1: Identify the inner and outer functions:

In this example, the inner function is g(x) = 2x + 1, and the outer function is f(x) = g(x)^3.

Step 2: Find the derivatives of the inner and outer functions individually:

The derivative of the inner function g(x) = 2x + 1 is dg(x)/dx = 2.

The derivative of the outer function f(x) = g(x)^3 is df(g(x))/dg(x) = 3(g(x))^2.

Step 3: Apply the chain rule formula:

Using the chain rule formula, we can find the derivative of f(x):

d(f(g(x)))/dx = (df(g(x))/dg(x)) * (dg(x)/dx)
= 3(g(x))^2 * 2
= 3(2x + 1)^2 * 2
= 6(2x + 1)^2

So, the derivative of f(x) = (2x + 1)^3 is 6(2x + 1)^2.

Remember, the chain rule is used when you have a composite function, and it allows you to find the derivative of the composite function by taking the derivative of the outer function with respect to the inner function and multiplying it with the derivative of the inner function with respect to x.

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