Question

How do you find `(dy)/(dx)` given `y^2tanx=x`?

Answer 1

`(dy)/(dx) = (1 - y^2sec^2x)/(2ytanx)`

Explanation:

We need to use the chain rule when differentiating a function of y since `y` itself is a function of `x`. First, differentiate that function of `y` with respect to `y`, then multiply by the derivative of `y` (with respect to `x`).

Then, we need to use the product rule: if `a,b` functions of `x`, then `(ab)' = a'b + ab'`.

Assuming `y` is a function of `x`, differentiate both sides:

`2y (dy)/(dx) tanx + y^2sec^2x = 1`

`(dy)/(dx) (2ytanx) = 1 - y^2sec^2x`

`(dy)/(dx) = (1 - y^2sec^2x)/(2ytanx)`