find d^2y/dx^2 of dy/dx = -[y/x]^1/3
To find the second derivative of y with respect to x, we can follow the steps given below:
Step 1: Start with the given equation:
dy/dx = -[y/x]^(1/3)
Step 2: Differentiate both sides of the equation with respect to x using the Chain Rule. Let's start by differentiating the left side:
d^2y/dx^2 = d/dx(dy/dx)
Step 3: Now, let's differentiate the right side of the equation. We need to apply the Chain Rule again since we have a composite function, [y/x]^(1/3). First, let's find the derivative of [y/x]^(1/3):
Let u = (y/x)
y = ux (multiply both sides by x)
dy/dx = u + x(du/dx) (apply the Product Rule)
dy/dx = u + x(du/dx)
dy/dx = (y/x) + x(du/dx)
dy/dx = (y + x(du/dx))/x
Now, let's differentiate the right side of the equation:
d^2y/dx^2 = d/dx[(y + x(du/dx))/x]
Step 4: Simplify this expression.
d^2y/dx^2 = d/dx[(y/x) + x(du/dx)]
d^2y/dx^2 = d/dx[(y/x)] + d/dx[x(du/dx)]
Step 5: Differentiate each term separately:
For the first term, d/dx[(y/x)], we can use the Quotient Rule:
d/dx[(y/x)] = [(x(dy/dx) - y(1))/x^2]
For the second term, d/dx[x(du/dx)], we apply the product rule:
d/dx[x(du/dx)] = x(d^2u/dx^2) + (du/dx)(dx/dx)
Since dx/dx is simply 1, this simplifies to:
d/dx[x(du/dx)] = x(d^2u/dx^2) + du/dx
Step 6: Substituting these values back into the expression:
d^2y/dx^2 = [(x(dy/dx) - y(1))/x^2] + x(d^2u/dx^2) + du/dx
Step 7: Replacing dy/dx and du/dx with their respective values from the original equation:
d^2y/dx^2 = [(x(-[y/x]^(1/3)) - y)/x^2] + x(d^2u/dx^2) + (-[y/x]^(1/3))
Finally, simplify the expression if needed. This is the second derivative of y with respect to x based on the given initial equation.