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

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