We are given the equation x^3 - x*y + y^2 = 17. We want to find the derivative y' = dy/dx.
Implicit differentiation allows us to find dy/dx when it may be difficult or impossible to express y explicitly in terms of x.
Now, let's apply the formula to each term.
The derivative of x^3 with respect to x will become 3x^2 using the power rule.
The derivative of x*y will become y + x*dy/dx by applying the product rule.
The derivative of y^2 with respect to x will become 2y*dy/dx by applying the chain rule.
The right hand side of equation 17 is a constant, so the derivative is zero.
Expressing this becomes:
3x^2 - (y + x*dy/dx) + 2y*dy/dx = 0
Next, we group the terms containing dy/dx on one side and move the terms without dy/dx to the other side:
dy/dx*(x - 2y) = y - 3x^2
At this point, since dy/dx stands for the derivative of y with respect to x, if we let y=x, then dy/dx=1. So let's substitute y with x in the above equation:
dy/dx*(x - 2x) = x - 3x^2
dy/dx*(-x) = - 2x^2
dy/dx = 2*x^2 / -x
This simplifies to:
dy/dx = -2x
However, don't forget, we made y=x in order to solve it, so the derivative should be in terms of x only, which is:
dy/dx = 3*x^2 - x
So, the derivative of the given function is 3x^2 - x.