How do you differentiate ln(cosh(ln(x) cos x)) and simplify?

Question

We haven't covered this in class, but i would like to know how to do it.

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Answer 1 · 2018-07-08T18:06:29

#f'(x)=-((ln(x)\sin(x)*x-cos(x))*sinh(ln(x)*cos(x)))/(x*cosh(ln(x)*cos(x))#

We use that

#(ln(x))'=1/x#

#(cosh(x))'=sinh(x)#
and

#(ln(x)*cos(x))'=1/x*cos(x)-ln(x)sin(x)#
so we get

#f'(x)=(sinh(ln(x)*cos(x))/(cosh(ln(x)*cos(x))))*(1/x*cos(x)-ln(x)*sin(x))#
which simplifies to

#f'(x)=-((ln(x)sin(x)*x-cos(x))*sinh(ln(x)*cos(x)))/(x*cosh(ln(x)*cos(x)))#