find dy/dx of sqrt(x+y)+sqrt(xy)=6
Use implicit differentiation and the "chain rule".
(1/2)(x+y)^-1/2*(1 + dy/dx) + (1/2)(xy)^-1/2*[x dy/dx + y] = 0
Solve for dy/dx. You can cancel out the "1/2" factors.
(x+y)^-1/2*(1 + dy/dx) + (xy)^-1/2*[x dy/dx + y] = 0
(x+y)^-1/2*dy/dx +(xy)^-1/2*[xdy/dx]
= -(x+y)^-1/2 -(xy)^-1/2*y
Take it from there
To find the derivative of the given equation, we will use implicit differentiation.
Step 1: Start by differentiating both sides of the equation with respect to x.
d/dx (sqrt(x+y) + sqrt(xy)) = d/dx (6)
Step 2: Apply the chain rule to differentiate the square root terms on the left side of the equation.
(1/2) * (x+y)^(-1/2) * (1 + (dy/dx)) + (1/2) * (xy)^(-1/2) * (y + x(dy/dx)) = 0
Step 3: Simplify the equation.
(1/2) * (1 + (dy/dx))/(sqrt(x+y)) + (1/2) * (y + xy(dy/dx))/(sqrt(xy)) = 0
Step 4: Multiply through by 2 to eliminate the denominators.
1 + (dy/dx)/(sqrt(x+y)) + (y + xy(dy/dx))/(sqrt(xy)) = 0
Step 5: Rearrange the terms to solve for dy/dx.
(dy/dx)/(sqrt(x+y)) + (y + xy(dy/dx))/(sqrt(xy)) = -1
Multiply through by (sqrt(x+y) * sqrt(xy)) to eliminate the denominators.
(dy/dx)(sqrt(xy)) + (y + xy(dy/dx))(sqrt(x+y)) = -sqrt(x+y) * sqrt(xy)
Step 6: Distribute and simplify.
dy/dx * sqrt(xy) + (y*sqrt(x+y) + xy*dy/dx*sqrt(x+y)) = -sqrt(x+y) * sqrt(xy)
Step 7: Combine like terms.
dy/dx * sqrt(xy) + xy*dy/dx*sqrt(x+y) = -sqrt(x+y) * sqrt(xy) - y*sqrt(x+y)
Step 8: Factor out dy/dx.
dy/dx * (sqrt(xy) + xy*sqrt(x+y)) = -sqrt(x+y) * sqrt(xy) - y*sqrt(x+y)
Step 9: Solve for dy/dx.
dy/dx = (-sqrt(x+y) * sqrt(xy) - y*sqrt(x+y))/(sqrt(xy) + xy*sqrt(x+y))
Hence, the derivative dy/dx of the given equation is (-sqrt(x+y) * sqrt(xy) - y*sqrt(x+y))/(sqrt(xy) + xy*sqrt(x+y)).