Find f' if x power 7 + y power 7 = 8xy

x^7 + y^7 = 8xy

7x^6 dx+7y^6 dy=8y dx + 8x dy

dy(7y^6-8x)=dx(8y-7x^6)

solve for dy/dx

x^7 + y^7 = 8xy

7x^6 + 7y^6 y' = 8y + 8x y'
8x y' - 7y^6 y' = 7x^6 - 8y
y'(8x - 7y^6) = 7x^6 - 8y
y' = (7x^6 - 8y)/(8x - 7y^6)

To find f', the derivative of f, we need to differentiate both sides of the equation with respect to x. Let's break down each term and differentiate them separately:

Term 1: x^7
Using the power rule for differentiation, the derivative of x^n with respect to x is n*x^(n-1).
So, the derivative of x^7 with respect to x is 7*x^(7-1) = 7x^6.

Term 2: y^7
Since we are differentiating with respect to x, y is considered a constant. So, the derivative of y^7 with respect to x is 0.

Term 3: 8xy
Using the product rule for differentiation, the derivative of a product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).
Here, u(x) = 8x and v(x) = y.
So, the derivative of 8xy with respect to x is 8y + 8x * dy/dx.

Now, let's sum up the derivatives of all the terms and set it equal to 0 since the derivative of a constant is 0:

7x^6 + 0 + 8y + 8x * dy/dx = 0.

Finally, we isolate the dy/dx term (the derivative of y with respect to x) to find f':
8x * dy/dx = -7x^6 - 8y.

Divide both sides by 8x:
dy/dx = (-7x^6 - 8y) / (8x).

Therefore, f'(x, y) = (-7x^6 - 8y) / (8x).