`y = (3(1 - sin(x)))/(2cos(x))` Find the derivative of the trigonometric function.

Textbook Question

Chapter 2, 2.3 - Problem 47 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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 You need to evaluate the derivative of the given function, using the quotient rule, such that:

`y' = ((3 - 3sin x)'*(2 cos x) - (3 - 3 sin x)*(2 cos x)')/(4 cos^2 x)`

`y' = (-3cos x*(2 cos x) + 2 sin x*(3 - 3 sin x))/(4 cos^2 x)`

`y' = (-6 cos^2 x+ 6sin x - 6 sin^2 x)/(4 cos^2 x)`

Factoring out -6 yields:

`y' = (-6 (cos^2 x - sin x + sin^2 x))/(4 cos^2 x)`

Using `sin^2 x + cos^2 x = 1 ` yields:

`y' = (-6/4)*(1 - sin x)/(cos^2 x)`

`y' = (-3/2)*(1 - sin x)/(cos^2 x)`

Hence, evaluating the derivative of the function yields

`y' = (-3/2)*(1 - sin x)/(cos^2 x).`

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