find the derivative of 5x^4y-3xy^6=6x^6(3y^4-7xy-8x^5)^10

Question

please show steps and reduce if possible, I missed school for this lesson today and am lost! thank you!

Answer 1

We will differentiate both sides of the equation using the chain rule.  This is known as implicit differentiation.  I will denote the derivative using single quotations (').

 

d/dx[5x⁴y - 3xy⁶] = d/dx[6x⁶(3y⁴ - 7xy - 8x⁵)¹⁰]

 

(20x³y + 5x⁴y') - (3y⁶ + 18xy⁵y') = (36x⁵(3y⁴ - 7xy - 8x⁵)¹⁰ + 60x⁶(3y⁴ - 7xy - 8x⁵)⁹(12y³y' - (7y + 7xy') - 40x⁴))

 

 

Distribute the terms and simplify wherever possible.  The goal is to get all of the terms with y' outside of parentheses.

 

 

20x³y + 5x⁴y' - 3x⁶ - 18xy⁵y' = 36x⁵(3y⁴ - 7xy - 8x⁵)¹⁰ + (3y⁴ - 7xy - 8x⁵)⁹(720x⁶y³y'- 420x⁶y - 420x⁷y' - 2400x¹⁰)

 

20x³y + 5x⁴y' - 3x⁶ - 18xy⁵y' = (3y⁴ - 7xy - 8x⁵)⁹(720x⁶y³y'- 420x⁶y - 420x⁷y' - 2400x¹⁰ + 36x⁵(3y⁴ - 7xy - 8x⁵))

 

20x³y + 5x⁴y' - 3x⁶ - 18xy⁵y' = (3y⁴ - 7xy - 8x⁵)⁹(720x⁶y³y' - 420x⁶y - 420x⁷y' - 2400x¹⁰ + 108x⁵y⁴ - 252x⁶y - 288x¹⁰)

 

 

The only way to get rid of the parentheses is to use the binomial expansion theorem on the right side. 

 

Then move all the y' terms to the left side and move all the non-y' terms to the right side.

Factor out the y' on the left side of equation. 

Divide both sides of the equation by the coefficient of y'.