The Chain Rule

Calculus Tutorial

The Chain Rule


How do we differentiate F(x)=Tan(2x-4)? This function is the composite FoG 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).


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)


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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