Solution to Exercise 6.3-1
© 1997 by Karl Hahn
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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