Karl's Calculus Tutor - Solution to Exercise 6.3-1

Solution to Exercise 6.3-1

The problem was to take the derivates of both ln(x²) and 2ln(x) and show they are the same. To find the derivative of the first one, observe that it is a composite, so you employ the chain rule. Let f(x) = ln(x) and g(x) = x². Then you are finding the derivative of f(g(x)).

From the discussion in this section you know that f'(x) = 1/x. From discussion in previous sections you know that g'(x) = 2x. The chain rule says to find the derivative of the composite, use f'(g(x)) * g'(x). Putting that together you have

    1          2
      × 2x  =   
   x²          x

To find the derivative of the second one, simply recall that when you need to find the derivative of a constant times a function, take the constant times the derivative of the function. So derivative of 2ln(x) is

   2
    
   x

and indeed the two derivatives are equal.

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