Chain rule facts for kids

The chain rule is a super useful tool in calculus that helps us find the derivative of a special kind of function. Imagine you have a function that's "inside" another function, like a set of Russian nesting dolls. This is called a composite function. The chain rule helps us figure out how fast the whole thing is changing.

Think of it like this: If you're riding a bike (the outer function) and the speed of your pedals (the inner function) affects how fast your bike goes, the chain rule helps you calculate your bike's speed based on how fast you're pedaling.

For example, if we have a function like F(x) = (x² + 5)³, the chain rule helps us find its derivative. It's used when one function is "wrapped around" another.

What is the Chain Rule?

The chain rule helps us find the derivative of a composite function. A composite function is when one function works on the result of another function. It looks like this:

• F(x) = f(g(x))

Here, 'g' is the "inside" function, and 'f' is the "outside" function. The chain rule tells us that the derivative of F(x), written as F'(x), is found by:

• F'(x) = f'(g(x)) * g'(x)

This means you take the derivative of the outside function (keeping the inside function as it is), and then you multiply that by the derivative of the inside function.

How to Use the Chain Rule

Using the chain rule involves a few simple steps:

Step 1: Find the Derivative of the Outside Function

• First, look at the "outer layer" of your function. Imagine the inside part is just a single variable.
• Find the derivative of this outside function. Don't change the inside part yet!

Step 2: Find the Derivative of the Inside Function

• Next, focus on the "inner layer" – the part of the function that's inside the brackets or parentheses.
• Find the derivative of this inside function.

Step 3: Multiply Them Together

• Finally, take the answer you got from Step 1 and multiply it by the answer you got from Step 2.
• This final product is the derivative of your original composite function!

Let's Look at an Example

Let's use the example: F(x) = (x² + 5)³

• In this function, the "outside" part is something cubed (something)³, and the "inside" part is (x² + 5).

• Step 1: Derivative of the outside function.

* Imagine the (x² + 5) is just 'u'. So you have u³. * The derivative of u³ is 3u². * Now, put the original inside part back in: 3(x² + 5)².

• Step 2: Derivative of the inside function.

* The inside function is (x² + 5). * The derivative of (x² + 5) is 2x (because the derivative of x² is 2x, and the derivative of 5 is 0).

• Step 3: Multiply the results.

* Multiply the result from Step 1 by the result from Step 2: * F'(x) = 3(x² + 5)² * (2x) * F'(x) = 6x(x² + 5)²

And that's how you use the chain rule! It helps you break down complex derivatives into smaller, easier-to-solve pieces.

Related pages

• Product rule

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