Karl's Calculus Tutor - Solution to Exercise 6.3-3

Solution to Exercise 6.3-3

The problem is to find an expression for the derivative of

   u(x)  =  ln(f(x))

You should have found this to be an easy application of the chain rule. If you had trouble with this, you should review the chain rule some more until you have a better understanding of it. The chain rule should be almost second nature to you by now.

You know this is a chain rule problem because it is asking for the derivative of a composite function. Let

   g(f)  =  ln(f)

It's asking you to take the derivative of u(x) = g(f(x)). The problem doesn't give expressions for f(x) or f'(x), so just use those symbols. You do know that

             1
   g'(f)  =   
             f

The chain rule tells you to find g'(f(x)) f'(x). Substituting the expression you have, you get

               1            f'(x)
   u'(x)  =       f'(x)  =       
             f(x)            f(x)

And that's the answer.

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