Essential Knowledge 2.1C4

Essential Knowledge 2.1C4 Students will know that the chain rule provides a way to differentiate composite functions.

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"Composite functions" is kind of a fancy word. For me, I just think of it as two functions mixed together in a way that isn't addition/ subtraction/ multiplication /division. You could also think of them as function inside of other functions. Example

f(x) = sin(2x³)

This is a composite function function because there is a 2x³ function inside of the sine function.

To solve these, one must follow a special rule called the chain rule. The chain rule has two forms.

Prime Notation Form of Chain Rule

The chain rule using prime notation is set up for working with functions (as opposed to equations). It looks like this:

For f(x) = g(h(x)), f'(x)= g'(h(x))·h'(x)

In other words, take the derivative of the outside function as if there were only an "x" inside, then multiply it times the derivative of the inside function.

Example

For f(x) = sin(2x³), find f'(x)

First we must determine what will be g and what will be h(x). This one is pretty obvious. g = sin and h(x) = 2x³. We'll also need the derivative of h(x), which would be h'(x) = 6x². Now we can use the formula.

f'(x) = cos(2x³)·6x² = 6x²cos(2x³)

Leibniz's Notation Form of Chain Rule

The chain rule using Leibniz's notation is set up for working with equations (as opposed to functions). It looks like this:

I really love this because it shows how the dy parts would cancel out. Let's see how this would play out in an example

Example

For q = sin(2n³), find q'

Let's change the q to a z and the n to an x temporarily just so our equations match the formula

z = sin(2x³), find z'

(Note that z' here is . You know this because x is the only other variable besides z.)

Now we must break this into an inside equation and an outside equation. The inside part is 2x³, so let's set that equal to y

y = 2x³

This means the our original equation becomes

z = sin(y)

We'll need to find the derivatives need for the formula

z = sin(y)

Now we can apply our chain rule formula

And just plug in y = 2x³ to get your final answer

And you can see we get the same answer using Leibniz's notation as we did using prime notation.