Question

Each of the equations in Exercises 69–80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find $\frac{dy}{dx}$

$4x^2-y^2=9$

Short answer

$\frac{dy}{dx}=\frac{4x}{y}$

Step-by-step solution

Step 1. Given Information:

Given equation: $4x^2-y^2=9$

We want to find $\frac{dy}{dx}$defines y as an implicit function of x by use implicit differentiation.

Step 2. Solution:

Differentiate both sides w.r.t. x

$$\begin{gathered} \frac{d}{dx}(4x^2-y^2)=\frac{d}{dx}\left(9\right) \\ \frac{d}{dx}4x^2-\frac{d}{dx}{y}^2=\frac{d}{dx}9 \\ 8x-2y·\frac{dy}{dx}=0 \\ 2y·\frac{dy}{dx}=8x \\ \frac{dy}{dx}=\frac{8x}{2y} \\ \frac{dy}{dx}=\frac{4x}{y} \end{gathered}$$