Find d^2y/dx^2 by implicit differentiation.
x^(1/3) + y^(1/3) = 4
I know that first you must find the 1st derivative & for y prime I got 1/3x^(-2/3) + 1/3y^(-2/3) dy/dx = 0
Then for dy/dx I got
dy/dx = [-1/3x^(-2/3)] / [1/3y^(-2/3)]
I think that from here I would use the quotient rule to find the second derivative?
simplify your first derivative before going further
notice you can divide each term by 1/3 to get
x^(-2/3) + y^(-2/3) dy/dx = 0
dy/dx = -x^(-2/3) / y^(-2/3)
= - y^(2/3)/x^(2/3)
= - (y/x)^(2/3)
d^2y/dx^2 = (-2/3) (y/x)^(-1/3) [( xdy/x - y)/x^2)
replace the dy/dx in the square bracket by (y/x)^2/3) and see what you get.
Still messy but a small improvement.
Yes, you are correct. To find the second derivative, you can use the quotient rule. Let's differentiate the expression you found for dy/dx using the quotient rule.
dy/dx = (-1/3x^(-2/3)) / (1/3y^(-2/3))
To apply the quotient rule, we differentiate the numerator and denominator separately and then use the following formula:
(d/dx)(u/v) = (v * du/dx - u * dv/dx) / v^2
Let's differentiate the numerator and denominator:
Numerator:
du/dx = d/dx(-1/3x^(-2/3))
To differentiate this expression, we need to apply the power rule and chain rule:
du/dx = -1/3 * d/dx(x^(-2/3))
Applying the power rule, we get:
du/dx = -1/3 * (-2/3) * x^(-2/3 - 1)
Simplifying the exponent, we have:
du/dx = 2/9 * x^(-5/3)
Denominator:
dv/dx = d/dx(1/3y^(-2/3))
To differentiate this expression, we need to apply the power rule and chain rule:
dv/dx = 1/3 * d/dx(y^(-2/3))
Applying the power rule and chain rule, we get:
dv/dx = 1/3 * (-2/3) * y^(-2/3 - 1) * dy/dx
Simplifying the exponent and substituting the value of dy/dx, we have:
dv/dx = (-2/9) * y^(-5/3) * (y^(-2/3)) * (1/3x^(-2/3))
Now we substitute the values obtained into the quotient rule formula:
(d/dx)(u/v) = (v * du/dx - u * dv/dx) / v^2
(d/dx)(dy/dx) = ((1/3y^(-2/3)) * (2/9 * x^(-5/3)) - (1/3x^(-2/3)) * (-2/9 * y^(-5/3) * y^(-2/3) * (1/3x^(-2/3)))) / (1/3y^(-2/3))^2
Simplifying the expression, we have:
(d/dx)(dy/dx) = (2/27 * x^(-5/3) * y^(-2/3) - 2/27 * y^(-7/3) * x^(-10/3)) / (1/27 * y^(-4/3))
Now, you can simplify this expression further if desired.