Q. 71 - 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 $\frac{dy}{dx}$

$x{y}^2+3x^2=4$

Step-by-Step Solution

Answer

$\frac{dy}{dx}=\frac{-y^2-6x}{2xy}$

Step 1. Given Information:

Given equation: $x{y}^2+3x^2=4$

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 

$$\begin{gathered} \frac{d}{dx}(x{y}^2+3x^2)=\frac{d}{dx}\left(4\right) \\ \frac{d}{dx}\left(x{y}^2\right)+\frac{d}{dx}\left(3x^2\right)=\frac{d}{dx}\left(4\right) \end{gathered}$$

Using Product and Chain Rule we get

$$\begin{gathered} (x\frac{d}{dx}{y}^2+y^2\frac{d}{dx}x)+6x=0 \\ x·2y·\frac{dy}{dx}+y^2+6x=0 \\ 2xy·\frac{dy}{dx}=-y^2-6x \\ \frac{dy}{dx}=\frac{-y^2-6x}{2xy} \end{gathered}$$