Solution to Exercise 6.3-1 © 1997 by Karl Hahn

---

The problem was to take the derivatives 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 × 2x = x2

2 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.

---

Return to Main Text

You can email me by clicking this button:

  Use your own emailer     Use form