Chain Rule

 

Chain Rule      If f and g are two differentiable functions, then

 

     By definition, . Let us say that g(x + h) = g(x) + u(x, h). (There is always a difference between g(x + h) and g(x) which depends on both x and h.)

 

     Replacing  with  in the definition of derivative, we have:

 

     .

    

     The only way the derivative can make sense is if  as .

 

     Using the definition of a derivative, we have . If we use the above technique and replace  with  we get:

 

     .

 

     At this point, a change of variables helps to simplify matters. Let  and let . Substituting these variables into the equation yields:

 

     .

 

     Using the same technique as we did for , notice that we can replace  with . Substituting this into the equation, we get:

 

     .

 

     Recall that  and . Plugging those values back in, we have:

 

    .