Question

In Exercises \(1-8,\) find \(d y / d x\). $$x=\tan y$$

Short answer

The derivative of the function \(x = \tan y\) with respect to x is \(dy/dx = \frac{1}{\sec^2 y}\).

Step-by-step solution

Rewrite

Rewrite the given equation as \(x - \tan y = 0\) for explicit implicit differentiation.

Implicit Differentiation

Differentiate implicitly by treating y as a function of x and using the chain rule for tan y. The chain rule for \( \tan y \) is \( \sec^2 y (dy/dx) \). So we get \(1 - \sec^2 y (dy/dx) = 0\).

Solve for \(dy/dx\)

Re-arrange the equation from step 2 to solve for \(dy/dx\). This gives us \(dy/dx = \frac{1}{\sec^2 y}\).