differentiate implicitly

Question

to find dy/dx if √x+8=2xy^2

Answer 1

√x + 8 = 2xy² 

 

y² = 1/(2√x) + 4/x

 

y = √(1/(2√x) + 4/x)

 

Differentiate via power rule & product rule

 

1/(2√x) = 2y² + 4xy*dy/dx

 

1/(2√x) - 2y² = 4xy*dy/dx

 

dy/dx = 1/(8x^3/2y) - y/2x

 

dy/dx = 1/(8x^3/2(√(1/(2√x) + 4/x) - √(1/(2√x) + 4/x)/2x

 

 

Answer 2

I am assuming the equation is   √(x + 8) = 2xy²

 

Rewrite the equation using exponents.

 

 

(x + 8)^1/2 = 2xy²

 

 

Derive both sides of the equation using the chain rule. 

 

(1/2)(x + 8)^-1/2 = 2y² + 2x*2yy'

 

(1/2)(x + 8)^-1/2 = 2y² + 4xyy'

 

 

Subtract 2y² on both sides of the equation.

 

-2y² + (1/2)(x + 8)^-1/2 = 4xyy'

 

 

Divide both sides of the equation by 4xy.

 

[-2y² + (1/2)(x + 8)^-1/2] / 4xy = y'