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Chain Rule
 
Question: What is a practical use of the Chain Rule?
 
Yours Truly answers:
The Chair Rule enables us to differentiate composite functions that were studied in precalculus.
 
Let  f  and  g  be fuctions. The composite function of  f  and  g, denoted by  f o g  (“f composite g”), is defined by  ( f o g )(x) = f (g(x)).
 
For example, if  f (x) = ex  and  g(x) = 3x, then

y = ( f o g )(x) = f (g(x)) = f (3x) = e3x.

At this stage, we assume that the only derivative rule we know for the exponential function is  (ex)' = ex. Let  u = 3x. Then, y = ( f o g )(x) = f (g(x)) = f (u) = eu. Thus,


This is the Chain Rule. Of course, we would normally write this computation as

(e3x)' = 3ex.

The key step is to let  u  replace the expression where normally  x  is located.
 
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