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 $\frac{dy}{dx}$

${(3y^2+5xy-2)}^4=2$

Short answer

$\frac{dy}{dx}=\frac{-5y}{(5x+6y)}$

Step-by-step solution

Step 1. Given Information:

Given equation:${(3y^2+5xy-2)}^4=2$

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

$\frac{d}{dx}{(3y^2+5xy-2)}^4=\frac{d}{dx}2$

Using product and chain rule we get

localid="1648700033816" $$\begin{gathered} 4{(3y^2+5xy-2)}^3\frac{d}{dx}(3y^2+5xy-2)=\frac{d}{dx}2 \\ 4{(3y^2+5xy-2)}^3\left(\frac{d}{dx}\left(3y^2\right)+\frac{d}{dx}\left(5xy\right)-\frac{d}{dx}\left(2\right)\right)=\frac{d}{dx}2 \\ 4{(3y^2+5xy-2)}^3\left(6y\frac{dy}{dx}+\left(5x\frac{dy}{dx}+5y\right)-0\right)=0 \\ (5x+6y)\frac{dy}{dx}+5y=0 \\ (5x+6y)\frac{dy}{dx}=-5y \\ \frac{dy}{dx}=\frac{-5y}{(5x+6y)} \end{gathered}$$