Post 4: Chain Rule

The chain rule is a fundamental rule in calculus that allows us to find the derivative of composite functions. A composite function is a combination of two or more functions, where the output of one function becomes the input of another. The chain rule helps us analyze how changes in the input for one function affect the output of the overall composite function.

Chain Rule Formula

The chain rule can be stated as follows:

If we have a composite function y = f(g(x)), where f is a function of g, and g is a function of x, then the derivative of y with respect to x is given by:

dy/dx = (dy/dg) * (dg/dx)

In other words, the derivative of the composite function is the derivative of the outer function multiplied by the derivative of the inner function.

Examples

Let's look at a couple of examples to understand how to apply the chain rule.

Example 1:

Consider the function y = (2x + 3)^4. We can see that this is a composite function, where the inner function is g(x) = 2x + 3, and the outer function is f(g) = g^4.

To find dy/dx using the chain rule, we need to find the derivatives of the inner and outer functions.

1. Step 1: Find the derivative of the inner function g(x): dg/dx = d(2x + 3)/dx = 2

2. Step 2: Find the derivative of the outer function f(g): df/dg = d(g^4)/dg = 4g^3

3. Step 3: Apply the chain rule formula: dy/dx = (dy/dg) * (dg/dx) = (4g^3) * 2 = 8(2x + 3)^3

So, the derivative dy/dx of the given function is 8(2x + 3)^3.

Example 2:

Let's consider another example with a different type of composite function. Suppose we have y = sin(x^2).

1. Step 1: Find the derivative of the inner function g(x): dg/dx = d(x^2)/dx = 2x

2. Step 2: Find the derivative of the outer function f(g): df/dg = d(sin(g))/dg = cos(g)

3. Step 3: Apply the chain rule formula: dy/dx = (dy/dg) * (dg/dx) = (cos(g)) * (2x) = 2x*cos(x^2)

So, the derivative dy/dx of the function y = sin(x^2) is 2x*cos(x^2).

The chain rule provides a powerful tool for finding the derivatives of complex functions built from simpler functions. By breaking down a composite function into its constituent parts and applying the chain rule, we can efficiently determine how changes in the input affect the overall function's rate of change.