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Imagine you have a machine that transforms a number (the first function), and its result is immediately fed into another machine to be processed again (the second function). This process is called a composite function. Mathematically, if we have as the inner function and as the outer function, their composition is .
The chain rule is a method for finding the derivative of functions like this, where one function is "nested" inside another.
To differentiate a composite function, we can't just differentiate it piece by piece. There's an elegant rule that connects the derivative of the outer function with the derivative of the inner function.
If , then its derivative is the product of the derivative of the outer function with respect to the inner function and the derivative of the inner function itself.
Formally, if we let , then . Its derivative is defined as:
This formula is known as the chain rule. Essentially, we differentiate from the outermost "layer" to the innermost layer and then multiply the results.
Let's try applying this theorem to a few cases to get a better feel for it.
Find the first derivative of .
Solution:
First, we need to break this function down into two parts: an outer function and an inner function.
Next, we find the derivative of each function:
Now, we can combine them using the chain rule:
Finally, we substitute back in and simplify the expression:
Find the first derivative of .
Solution:
We can rewrite this function as . This is a case where we need to apply the chain rule more than once because there are three layers of functions.
Then, we differentiate each layer:
Combine them all using the chain rule:
Now, substitute and back in step-by-step to get the final result:
Solution for
Step : Identify the functions
Step : Find the derivative of each
Step : Combine with the chain rule
Step : Don't forget to substitute back
Solution for
This function is . We'll use the multi-layered chain rule.
Step : Identify the functions
Step : Find the derivative of each
Step : Combine with the chain rule
Step : Substitute back and simplify
