Differentiate x^2+y^3=27xy

7/y-27

use implicit differentiation. Don't forget the chain rule and product rule. You have

2x + 3y^2 y' = 27y + 27xy'
The rest is just algebra: solve for y'

To differentiate the equation x^2 + y^3 = 27xy, we will need to find the derivatives of both sides with respect to x. This will involve using the rules of differentiation for terms involving x and y.

First, let's find the derivative of x^2 with respect to x. The power rule states that if we have a term x^n, the derivative with respect to x is given by nx^(n-1). Applying this rule, we get:

d/dx (x^2) = 2x

Next, let's find the derivative of y^3 with respect to x. Since y is a function of x, we will need to use the chain rule. The chain rule states that if we have a function f(g(x)), then the derivative with respect to x is given by f'(g(x)) * g'(x). In our case, y is the function g(x) and y^3 is f(y). Applying the chain rule, we get:

d/dx (y^3) = 3y^2 * dy/dx

To find dy/dx, we need to differentiate y with respect to x. However, to do this we need to know the relationship between x and y. The equation x^2 + y^3 = 27xy does not explicitly give the relationship between x and y, so we cannot differentiate y with respect to x.

Therefore, we are unable to find the derivative of the equation x^2 + y^3 = 27xy without additional information about the relationship between x and y.