find d^2y/dx^2 of x^2/3 + y^2/3 =1
(implicit differentiation)
Differentiate both sides with respect to x, with y considered a function of x.
(2/3)/x^1/3 + [(2/3)/y^1/3] dy/dx = 0
dy/dx = -[y/x]^1/3
Then do it again.
d^2/dx^2 = .......
To find the second derivative of y (d²y/dx²) for the given equation using implicit differentiation, we need to differentiate both sides of the equation with respect to x twice. Here's how you can do it step by step:
Step 1: Start with the original equation:
x^2/3 + y^2/3 = 1
Step 2: Differentiate both sides of the equation with respect to x:
(d/dx) (x^2/3) + (d/dx) (y^2/3) = (d/dx) (1)
The derivative of 1 with respect to x is zero, so we can simplify the equation to:
(2/3)x^(-1/3) + (2/3)y^(-1/3) * (dy/dx) = 0
Step 3: Now, differentiate both sides of the equation again with respect to x. The product rule will be used for the second term on the left side, since we have y as a function of x.
Using the product rule, (d/dx) (y^(-1/3)*(dy/dx)) equals:
(dy/dx)*(-1/3)*y^(-4/3) + y^(-1/3)*((d^2y/dx^2))
So the equation becomes:
-(2/9)x^(-4/3) + (2/3)*(-1/3)*y^(-4/3)*(dy/dx) + y^(-1/3)*((d^2y/dx^2)) = 0
Step 4: Rearrange the equation to solve for (d^2y/dx^2):
y^(-1/3)*((d^2y/dx^2)) = (2/9)x^(-4/3) - (2/9*y^(-4/3))*(dy/dx)
Finally, we have the second derivative formula for y in terms of x and y:
(d^2y/dx^2) = [(2/9)x^(-4/3) - (2/9*y^(-4/3))*(dy/dx)] / y^(-1/3)
So, to find the second derivative (d^2y/dx^2) at a specific point or evaluate it for a given x and y value, substitute the values into this formula.