Use implicit differentiation to find the slope of the tangent line to the curve 4xy^3+5xy=27 Point:(3,1)

Rhett Guerrero

Rhett Guerrero

Answered question

2022-11-14

4xy^3+5xy=27 Point:(3,1)
So, using Implicit Differentiation...
(4y^3+4xy^2(dy/dx))+(5y+5x(dy/dx))=0
4xy^2(dy/dx)+5x(dy/dx)=-4y^3-5y
dy/dx(4xy^2-5x)=-4y^3-5y
dy/dx=(-4y^3-5y)/(4xy^2+5x)
Now I replace x and y in the equation with x=3 and y=1
I get 20/27, which is apparently incorrect. What am I doing wrong?

Answer & Explanation

mignonechatte00f

mignonechatte00f

Beginner2022-11-15Added 13 answers

If you perform the differentiation, the first step leads to
( 4 y 3 + 12 x y 2 d y d x ) + ( 5 y + 5 d y d x ) = 0
which leads to
d y d x = 4 y 3 + 5 y x ( 12 y 2 + 5 )
and at (3,1) the value is the 3 17 .
Alberto Calhoun

Alberto Calhoun

Beginner2022-11-16Added 5 answers

When you differentiate y^3 you have to lower the 3 as a coefficient

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