Given that $3x-tany=4$, what is $\frac{dy}{dx}$ in terms of $y$?

• A. $3{cosy}^2$
• B. $\frac{1}{3{cosy}^2}$
• C. $\frac{1}{siny}$
• D. $-3{cosy}^2$

Answer: (A)

Verified step by step guidance

1. Start by differentiating both sides of the given equation, 3x - tan(y) = 4, with respect to x. Remember that y is a function of x, so you will need to apply the chain rule when differentiating tan(y).
2. The derivative of 3x with respect to x is 3. The derivative of -tan(y) with respect to x is -sec^2(y) * dy/dx (using the chain rule). The derivative of the constant 4 is 0.
3. Combine the results from the differentiation step to form the equation: 3 - sec^2(y) * dy/dx = 0.
4. Solve for dy/dx by isolating it on one side of the equation. Rearrange to get dy/dx = 3 / sec^2(y).
5. Simplify the expression for dy/dx using the trigonometric identity sec^2(y) = 1 / cos^2(y). This gives dy/dx = 3 * cos^2(y).