Question

Find dy/dx implicitly and find the largest interval of the form −a < y < a or 0 < y < a such that y is a differentiable function of x.

Answer 1

Let's find the implicit derivative:

$\begin{gathered} \frac{d}{dx}\tan ^2y=\frac{d}{dx}(x) \\ \sec ^2y\frac{dy}{dx}=1 \\ \frac{dy}{dx}=\frac{1}{\sec^2y} \\ \frac{dy}{dx}=\cos ^2y \end{gathered}$

Therefore:

$\frac{dy}{dx}=\cos ^2y$

Since the derivative of y is equal to the cosine and the cosine is a continous function for all the real numbers then the derivative of y is defined for all values of y.