The Chain Rule

Chain rule question

Find the derivative of this function with respect to x:

Solution

Note that this function can be split into two functions:

If we let that sub-function = u, or any letter we want, we can rewrite the function as:

                                                             y = 4u⁵

                                                      where u = 3x + 4

All the chain rule says is that

In words, this says the derivative of y with respect to x is equal to the derivative of y with respect to u times the derivative of u with respect to x.  Note if you cancel out the ‘du’s on the right hand side, you are left with the left hand side.

Now:

And:

So we can use our differentiation rules to calculate that

We can then substitute in for what u is and give the answer as

                                                          60(3x + 4)⁴

Chain rule question

Find the derivative of this function with respect to x:

Solution

Let u = 3x² + 2x + 3.

Then the function becomes y = cos u.

Now:

Let’s work out what each bit equals:

And put it all together: