Q. 72 - 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 $\frac{dy}{dx}$

$y^6-3x+4=0$

Step-by-Step Solution

Answer

$\frac{dy}{dx}=\frac{1}{2y}$

Step 1. Given Information:

Given equation: $y^6-3x+4=0$

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

$$\begin{gathered} \frac{d}{dx}(y^6-3x+4)=\frac{d}{dx}0 \\ \frac{d}{dx}\left(y^6\right)-\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(4\right)=\frac{d}{dx}0 \\ 6y·\frac{dy}{dx}-3+0=0 \\ 6y·\frac{dy}{dx}=3 \\ \frac{dy}{dx}=\frac{3}{6y} \\ \frac{dy}{dx}=\frac{1}{2y} \end{gathered}$$