## Calculus Tutorial

*The Chain Rule*

#### Intro

How do we differentiate F(x)=Tan(2x-4)? This function is the composite *F*o*G* of two functions *F* (u)=Tan u and u=2x-4 that we know how to differentiate. The *Chain rule* states that the correct answer to F(x) is derived by multiplying the derivatives of *F* and *G*.

#### Sample Problem

Find the derivative of F(x)=Tan(2x-4).

#### Solution

To begin, it is best to start by breaking the problem into its *F* (u) or

*F* form and its u or *G* forms.

*F* (u)=*F*=Tan(u)

u=*G*=2x-4

Now that we’ve done that, we should take the derivatives of each form.

*F* ‘(u)=*F* ‘=Sec^2(u)

(Notice the u in the parenthesis is **NOT** derived)

u ‘=*G* ‘= 2

Using the Chain Rule, we know that we must now multiply *F* ‘ and *G* ‘.

F ‘(x)=2*Sec^2(u)

We cannot leave u in our final answer, so knowing that u=2x-4, we can write the final answer as:

F ‘(x)=2*Sec^2(2x-4)

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