What is the derivative of #f(x) = ln(x/5)#?

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Answer 1 · 2015-10-28T04:38:21

#1/x#

We need to use the chain rule to solve this. The chain rule states that if you have a function #f(x)# where;

#f(x)=g(h(x))#

Then;

#f'(x)=g'(h(x))h'(x)#

The given function breaks down to;

#g(h)=ln(h)#
#h(x)=x/5#

Now we need to find the derivatives of #g# and #h#. For a proof on the derivative of #ln(x)#, look here .

#g'(h)=1/h#
#h'(x)=1/5#

Now applying the chain rule;

#f'(x)=g'(h(x))h'(x)=1/(x/5)(1/5)=(cancel5/x)(1/cancel5)=1/x#