Suppose that x and y are related by the equation x^2/4 + y^3/2 = 4. Use implicit differentiation to determine dy/dx.
My answer was 1/3y^2, y cannot = 0 but this is wrong.
x/2+3/2 y^2 y'=0
solve for y'
y'=-x/3y^2
check me.
To find dy/dx using implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Let's start by differentiating the left side of the equation:
d/dx (x^2/4 + y^3/2)
To differentiate x^2/4 with respect to x, we can treat x as the variable and apply the power rule. The power rule states that if we have a term of the form x^n, the derivative with respect to x is n*x^(n-1).
Therefore, the derivative of x^2/4 with respect to x is (2/4)x^(2-1) = (1/2)x.
To differentiate y^3/2 with respect to x, we need to apply the chain rule. The chain rule states that if we have a composite function, y = f(g(x)), then the derivative of y with respect to x is dy/dx = (df/dg) * (dg/dx).
In this case, g(x) = y, so dg/dx = dy/dx.
Using the power rule, df/dg = (3/2)y^(3/2 - 1) = (3/2)y^(1/2).
Therefore, the derivative of y^3/2 with respect to x is (3/2)y^(1/2) * dy/dx.
Now, let's differentiate the right side of the equation:
d/dx (4) = 0 (since 4 is a constant)
Now, we can set up our equation:
(1/2)x + (3/2)y^(1/2) * dy/dx = 0
Now, we can isolate dy/dx to solve for it:
(3/2)y^(1/2) * dy/dx = -(1/2)x
dy/dx = -(1/2)x / ((3/2)y^(1/2))
Simplifying further:
dy/dx = -x / (3y^(1/2))
So, the correct answer for dy/dx is -x / (3y^(1/2)). This is different from your answer of 1/3y^2.