Question

Find dy/dx by implicit differentiation.
√xy=−5−4x^2 *y

dy/dx=_____?_______

Answer 1

To find dy/dx, we differentiate both sides of the equation √(xy) = -5-4x^2y with respect to x using the chain rule. Simplify the equation and solve for dy/dx to find the derivative.

Step-by-step explanation:

To find ∬(dy/dx) in terms of x and y, we will differentiate both sides of the equation √(xy) = -5-4x^2y with respect to x.

Using the chain rule, we get:

• (1/2)(xy)^(-1/2)(y + x(dy/dx)) = -8x(2y) - 4x^2(dy/dx)

Simplifying the equation, we get:

• (y + xy(dy/dx))/(2√(xy)) = -16xy - 4x^2(dy/dx)

Now, we can solve for dy/dx:

• (y + xy(dy/dx))/(2√(xy)) + 4x^2(dy/dx) = -16xy
• (y + xy(dy/dx)) + 2x^2(dy/dx)√(xy) = -32xy(√(xy))
• (y + xy(dy/dx)) + 2x^2(dy/dx)√(xy) = -32xy(√(xy))
• 
  
• dy/dx = (-32xy(√(xy)) - y)/(x(2x √(xy) + y))