find d^2y/dx^2 of dy/dx = -[y/x]^1/3
drwls gave you the first half of the solution.
where are you getting stuck in the 2nd part?
To find the second derivative of y with respect to x, we'll need to apply the chain rule and differentiate both sides of the given equation. Let's go step by step:
Step 1: Rewrite the equation
The given equation is:
dy/dx = -[y/x]^(1/3)
Step 2: Apply the chain rule
Let's define u = y/x. The equation becomes:
dy/dx = -u^(1/3)
Now, we can differentiate both sides of the equation with respect to x:
Step 3: Differentiate dy/dx
Differentiating dy/dx with respect to x gives us the second derivative:
d^2y/dx^2
Step 4: Differentiate on the left side
Since dy/dx is the first derivative of y with respect to x, we can differentiate it again:
d/dx (dy/dx) = d/dx (-u^(1/3))
Step 5: Apply the chain rule on the right side
To differentiate -u^(1/3), we have a composition of functions. We need to differentiate the outer function, -u^(1/3), and also the inner function, u = y/x.
The derivative of -u^(1/3) with respect to u is:
d/du (-u^(1/3)) = -(1/3)u^(-2/3)
However, remember that we need to differentiate with respect to x, so we'll multiply this by du/dx.
Step 6: Compute du/dx
Differentiating u = y/x with respect to x gives us:
du/dx = (d/dx)y/x
To differentiate y/x, we can use the quotient rule. The numerator is y, and the denominator is x. Applying the quotient rule gives us:
dy/dx * x - y * 1 = x(dy/dx) - y
Now, we can substitute du/dx back into our equation:
-(1/3)u^(-2/3) * du/dx = -(1/3)u^(-2/3) * (x(dy/dx) - y)
Step 7: Substitute back y/x for u
Recall that we defined u = y/x. So, we can substitute u back into our equation:
-(1/3)(y/x)^(-2/3) * (x(dy/dx) - y)
This is the second derivative, d^2y/dx^2, in terms of y and x.
Note: The expression can be simplified further if needed by combining terms and performing any necessary algebraic simplifications.