Find $\frac{dy}{dx}$ of $\tan(2x+y) = 2x$.
All of the possible answer choices do not include a trigonometric function in them, so I figured there might be some substitution required.
So far, the process I took was to differentiate both sides and use the chain rule for the left side.
$$\sec^2(2x+y)(2 + y') = 2$$
$$2 + y' = \frac{2}{\sec^2(2x+y)}$$
$$y' = \frac{2}{\sec^2(2x+y)} – 2$$
$$y' = \frac{2}{\sec^2(2x+y)} – \frac{2\sec^2(2x+y)}{\sec^2(2x+y)}$$
$$y' = \frac{2(1-\sec^2(2x+y))}{\sec^2(2x+y)}$$
$$y' = \frac{-2\tan^2(2x+y)}{\sec^2(2x+y)}$$
$$y' = -2\sin^2(2x+y)$$
I'm stuck here because I'm not sure if there are any substitutions to make here to remove the trigonometric function from the answer.
Best Answer
$y' = \frac{2}{\sec^2(2x+y)} - 2$
$y' = \frac{2}{1+\tan^2(2x+y)} - 2$
And now we substitute from the original!
$tan(2x+y) = 2x$
$y' = \frac{2}{1+4x^2} - 2$
$y' = \frac{2-2-8x^2}{1+4x^2}$
$y' = \frac{-8x^2}{1+4x^2}$