Question

Find the slope of the curve 2(x+y)^1/3=y at the point (4,4)

Answer 1

Take the derivative.
y^3 - xy^2 = 4
3y^2(dy/dx) - (y^2 + 2xy(dy/dx)) = 0

Isolate dy/dx.
3y^2(dy/dx) - 2xy(dy/dx) = y^2, factor dy/dx
(dy/dx)(3y^2 - 2xy) = y^2, divide both sides by (3y^2 - 2xy)
dy/dx = y^2/(3y^2 - 2xy) is your derivative equation.

Now we need x. Substitute 2 for y in the original equation and solve for x.
(2)^3 - x(2)^2 = 4
4x = 8 - 4
4x = 4
x = 1

Now, substitute x and y into the derivative's equation.
dy/dx = (2)^2/(3(2)^2 - 2(1)(2)) = 4/(12 - 4) = 1/2

The slope, dy/dx, is 1/2 when y = 2.

Answer 2

there ya go

Answer 3

wow thanks

Answer 4

but this wasnt my problem

Answer 5

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