Q. 70 - Calculus

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 $\frac{dy}{dx}$

$x^2+y^2=4$

Step-by-Step Solution

Answer

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

Step 1. Given Information:

Given equation: $x^2+y^2=4$

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

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