## The Chain Rule

The best way to demonstrate this is by example.

 Sponsored Links 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

Here’s another example question with trigonometric ratios.

 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: