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#