Question

Find the derivative of the transcendental function. $$ y=\frac{3(1-\sin x)}{2 \cos x} $$

Short answer

The derivative of the function \(y=\frac{3(1-\sin x)}{2 \cos x}\) is \(y'=\frac{3(2-\sin x)}{2 \cos^2 x}\).

Step-by-step solution

Quotient Rule Preparation

Differentiate the numerator and the denominator separately. The derivative of \(3(1-\sin x)\) using the chain rule results in -3\(\cos x\). The derivative of \(2 \cos x\) using the basic derivative rule results in -2\(\sin x\).

Apply Quotient Rule

According to the quotient rule, \((f/g)'\) is equal to \((g f'-f g') / g^2\). Applying this rule to our function, we have \(y' = \frac{-2\sin x \cdot -3\cos x - -2\sin x \cdot 3(1-\sin x)}{(2\cos x)^2}\). Simplify this expression by multiplying and combining like terms.

Simplify the Expression

The simplified version of the derivative is \(\frac{6\sin^2 x + 6 \cos^2 x - 6\sin x}{4\cos^2 x}\). Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to further simplify.

Further Simplification Using Trigonometric Identity

Applying the identity results in \(\frac{6 - 6\sin x}{4\cos^2 x}\) or \( \frac{3(2-\sin x)}{2 \cos^2 x}\).