What is chain rule?

The Chain Rule is a fundamental concept in calculus that allows us to differentiate composite functions. In simpler terms, it helps us find the derivative of a function that is inside another function.

Here's the basic idea:

If we have a composite function f(g(x)), its derivative with respect to x is given by:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

Where:

• f'(g(x)) is the derivative of the outer function f evaluated at the inner function g(x).
• g'(x) is the derivative of the inner function g(x).

In essence, the chain rule states that the derivative of a composite function is the product of the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function.

Here are some key aspects and applications of the chain rule:

• Identifying Composite Functions: The first step in applying the chain rule is to correctly identify the outer and inner functions.
• Repeated Application: Sometimes, you might encounter nested composite functions (e.g., f(g(h(x)))). In such cases, you need to apply the chain rule multiple times.
• Implicit Differentiation: The chain rule is also vital in implicit%20differentiation, where you differentiate equations that are not explicitly solved for one variable in terms of the other.
• Related Rates Problems: Many related%20rates problems, which involve finding the rate of change of one quantity in terms of the rate of change of another, require the use of the chain rule.

Example:

Let's say y = sin(x^2).

1. Identify Outer and Inner Functions:

   • Outer function: f(u) = sin(u)
   • Inner function: g(x) = x^2

2. Find Derivatives:

   • f'(u) = cos(u)
   • g'(x) = 2x

3. Apply the Chain Rule:

   dy/dx = f'(g(x)) * g'(x) = cos(x^2) * 2x = 2x * cos(x^2)