`y = int_(cos(x))^(sin(x))(ln(1 + 2v))dv` Find the derivative of the function.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the the derivative of the function, hence, you need to use the part1 of fundamental theorem of calculus:

`y = int_a^b f(x)dx => (dy)/(dx) = f(x)` for `x in (a,b)`

If `f(x) = ln(1+2x)` , yields:

`(dy)/(dx) = ln(1+2x)|_(cos x)^(sin x)`

`(dy)/(dx) = ln(1+2sin x) - ln(1+2cos x)`

Using the properties of logarithms, yields:

`(dy)/(dx) = ln ((1+2sin x)/(1+2cos x))`

Hence, evaluating the derivative of the function, using the part 1 of the fundamental theorem of calculus, yields `(dy)/(dx) = ln ((1+2sin x)/(1+2cos x))`