Find dy/dx by implicit differentiation

Question

Find dy/dx by implicit differentiation

 

(9x + y)^1/2 = 3 +x²y²

Answer 1

We have to use the power rule plus the chain rule on the left side of the equation, and the product rule, power rule, and chain rule on the right side.

 

We get  (1/2)(9x+y)^-1/2(9+y') = 0 + 2xy² + x²(2y)y'

 

(9+y')/[2(9x+y)^1/2 = 2xy² + 2xyy'

 

Now multiply both sides by the denominator of the left side, to clear the fraction:

 

9+y' = 2(9x+y)^1/2(2xy²) + 2(9x+y)^1/2(2xyy')

 

Now get the terms with y' over to the left side, and the 9 over to the right side:

 

y' - 2(9x+y)^1/2(2xyy') = 4xy²(9x+y)^1/2 - 9

 

Every term on the left side has a y' in it, so factor y' out of the left side:

 

y'[1 - 4xy(9x+y)^1/2] = 4xy²(9x+y)^1/2 - 9

 

Divide both sides by the piece in the brackets on the left side, leaving just y' behind:

 

 

y' = [4xy²(9x+y)^1/2 - 9]/[1 - 4xy(9x+y)^1/2]

 

Wasn't that fun?