Q. 76 - 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 $\frac{dy}{dx}$

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

Step-by-Step Solution

Answer

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

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 

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