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The Chain Rule in Calculus: A Step-by-Step Guide
In calculus, the chain rule is a fundamental concept that allows us to find the derivative of a composite function. A composite function is a function that is made up of two or more functions, where the output of one function becomes the input of another function.
For example, consider the function f(x) = sin(x^2). This function is a composite function because it combines the sine function (sin) and the squaring function (x^2). To find the derivative of this composite function, we use the chain rule.
What is the Chain Rule?
The chain rule states that the derivative of a composite function is equal to the product of the derivative of the outer function and the derivative of the inner function.
Mathematically, this can be expressed as:
d/dx [f(g(x))] = f'(g(x)) * g'(x)
where:
- f(x) is the outer function
- g(x) is the inner function
- f'(x) is the derivative of the outer function
- g'(x) is the derivative of the inner function
Example: Finding the Derivative of a Composite Function
Let’s find the derivative of the function f(x) = sin(x^2) using the chain rule.
1. **Identify the outer and inner functions.**
- Outer function: f(u) = sin(u)
- Inner function: g(x) = x^2
2. **Find the derivatives of the outer and inner functions.**
- f'(u) = cos(u)
- g'(x) = 2x
3. **Substitute the inner function into the derivative of the outer function.**
f'(g(x)) = cos(x^2)
4. **Multiply the result by the derivative of the inner function.**
d/dx [f(g(x))] = cos(x^2) * 2x = 2x cos(x^2)
Therefore, the derivative of f(x) = sin(x^2) is 2x cos(x^2).
Key Points to Remember
- The chain rule is used to find the derivative of a composite function.
- The derivative of the composite function is equal to the product of the derivative of the outer function and the derivative of the inner function.
- When applying the chain rule, remember to substitute the inner function into the derivative of the outer function.
Practice Problems
Here are some practice problems to test your understanding of the chain rule:
- Find the derivative of f(x) = (x^2 + 1)^3.
- Find the derivative of g(x) = cos(2x).
- Find the derivative of h(x) = sqrt(x^3 + 1).
The chain rule is a powerful tool in calculus that helps us understand the behavior of composite functions. With practice, you’ll be able to apply this rule confidently to solve a wide range of problems.