Use implicit differentiation to find
dy/dx if
384,000=30x^1/3y^2/3 .
384000 = 30 x^1/3 y^2/3
y^-2/3 = 1/12800 x^1/3
-2/3 y^-5/3 y' = 1/38400 x^-2/3
y' = -y/25600 (y^2/x^2)^(1/3)
thank you very much for your help
To find dy/dx using implicit differentiation, we'll differentiate both sides of the equation with respect to x. Remember that when differentiating y with respect to x, we use the chain rule which states that the derivative of y with respect to x is (dy/dx).
Given the equation:
384,000 = 30x^(1/3)y^(2/3)
Step 1: Differentiate both sides of the equation with respect to x.
On the left-hand side, the derivative of a constant is zero.
On the right-hand side, we need to differentiate both terms separately using the chain rule.
For the first term, 30x^(1/3), we have:
d(30x^(1/3))/dx = (1/3) * 30 * x^(-2/3) * dx/dx
=> 10x^(-2/3)
For the second term, y^(2/3), we have:
d(y^(2/3))/dx = (2/3) * y^(-1/3) * (dy/dx)
Step 2: Rearrange the equation to solve for (dy/dx).
384,000 = 10x^(-2/3) * y^(2/3) * (dy/dx)
Step 3: Solve for (dy/dx).
To isolate (dy/dx), divide both sides of the equation by 10x^(-2/3) * y^(2/3):
(dy/dx) = 384,000 / (10x^(-2/3) * y^(2/3))
Simplifying further, we can rewrite the equation as:
(dy/dx) = 38,400 * x^(2/3) / y^(2/3)
Thus, the derivative of y with respect to x, dy/dx, is 38,400 times x raised to the power of 2/3, divided by y raised to the power of 2/3.