What is the derivative of $f(x)=ln(x/sinx)$ ?

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Answer 1 · 2018-05-13T21:42:58

$d/dx (ln(x/sinx)) = 1/x-cotx$

Using the chain rule:

$d/dx (ln(x/sinx)) = 1/(x/sinx) d/dx (x/sinx)$

now using the quotient rule:

$d/dx (ln(x/sinx)) = sinx/x (sinx-xcosx)/sin^2x$

and simplifying:

$d/dx (ln(x/sinx)) = (sinx-xcosx)/(xsinx)$

$d/dx (ln(x/sinx)) = 1/x-cotx$

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