How to use chain rule in calculus?

The chain rule is a fundamental rule in calculus that allows you to differentiate composite functions. It is used when you have a function within a function, and you need to find the derivative of the composite function.

Here is the general formula for the chain rule:

If y = f(g(x)), where f(u) and g(x) are differentiable functions, then the derivative of y with respect to x is given by:

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

In other words, the derivative of the outer function (f(u)) with respect to the inner function (u) multiplied by the derivative of the inner function (g(x)) with respect to x.

To use the chain rule, follow these steps:

1. Identify the outer function (f(u)) and the inner function (g(x)) in the composite function.

2. Differentiate the outer function (f(u)) with respect to the inner function (u). This means you treat u as the variable and differentiate f(u) as you normally would.

3. Differentiate the inner function (g(x)) with respect to x. This means you treat x as the variable and differentiate g(x) as you normally would.

4. Multiply the results from steps 2 and 3 together to find the derivative of the composite function.

5. Simplify the result if possible.

Here is an example to illustrate the use of the chain rule:

Given y = (2x^3 + 1)^4, find dy/dx.

1. Identify the outer function (f(u)) as u^4 and the inner function (g(x)) as 2x^3 + 1.

2. Differentiate the outer function (f(u)) with respect to the inner function (u):
df/du = 4u^3.

3. Differentiate the inner function (g(x)) with respect to x:
dg/dx = 6x^2.

4. Multiply the results from steps 2 and 3 together:
dy/dx = (4u^3) * (6x^2) = 24u^3x^2.

5. Simplify the result:
dy/dx = 24(2x^3 + 1)^3 * x^2.

So, the derivative of y = (2x^3 + 1)^4 with respect to x is dy/dx = 24(2x^3 + 1)^3 * x^2.