Find d2y/dx2 in terms of x and y.
y5 = x6
To find the second derivative of y (d^2y/dx^2) in terms of x and y, we need to differentiate the equation y^5 = x^6 twice.
First, let's differentiate the given equation y^5 = x^6 implicitly with respect to x:
Taking the derivative of both sides with respect to x:
(d/dx) (y^5) = (d/dx) (x^6)
To differentiate y^5, we use the chain rule:
5y^4 (dy/dx) = 6x^5
Next, we can rearrange the equation to solve for (dy/dx):
(dy/dx) = (6x^5) / (5y^4)
Now, let's differentiate the equation (dy/dx) = (6x^5) / (5y^4) implicitly with respect to x to find the second derivative:
Taking the derivative of both sides with respect to x:
(d^2y/dx^2) = d/dx [(6x^5) / (5y^4)]
Using the quotient rule, we can differentiate the right-hand side of the equation:
(d^2y/dx^2) = (5y^4 d/dx(6x^5) - 6x^5 d/dx(5y^4)) / (5y^4)^2
Next, let's differentiate the terms within the numerator:
(d^2y/dx^2) = (5y^4 * 30x^4 - 6x^5 * 20y^3) / (5y^4)^2
Simplifying the expression:
(d^2y/dx^2) = (30x^4y^4 - 120x^5y^3) / 25y^8
Therefore, the second derivative of y (d^2y/dx^2) in terms of x and y is:
(d^2y/dx^2) = (30x^4y^4 - 120x^5y^3) / 25y^8