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} (d)/(dx)\tan ^2y=(d)/(dx)(x) \\ \sec ^2y(dy)/(dx)=1 \\ (dy)/(dx)=(1)/(\sec^2y) \\ (dy)/(dx)=\cos ^2y \end{gathered}`

Therefore:

`(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.