Unusual partial differentiation equation

[rachibabes]

Homework Statement

Calculate ∂f/∂x and ∂f/∂y for the following function:

yf^2 + sin(xy) = f

The Attempt at a Solution

I understand basic partial differentiation, but I have no idea how to approach the f incorporation on both sides of the equation nor what you would explicitly call this kind of mathematical technique. Anyone who can point me in the right direction?

 

[tiny-tim]

welcome to pf!

hi rachibabes! welcome to pf!

Calculate ∂f/∂x and ∂f/∂y for the following function:

yf^2 + sin(xy) = f

just do ∂/∂x to the equation in the usual way (using the product rule for the yf²) …

what do you get? ​

 

[rachibabes]

yf^2 + sin(xy) = f

I get:

y2f∂f/∂x +f^2∂y/∂x + cos(xy)*(x∂y/∂x+y) = ∂f/∂x

y2f∂f/∂x +f^2∂y/∂x + x∂y/∂xcos(xy) + ycos(xy) = ∂f/∂x

∂y/∂x[f^2 + cos(xy)] + ycos(xy) + y2f∂f/∂x = ∂f/∂x

I have no idea how to remove the ∂f/∂x on the left hand side :/

 

[tiny-tim]

hey there, rachibabes!

(just got up :zzz:)

f(x,y) is a function of the variables x and y

∂/∂x means differentiating wrt x keeping y fixed

so ∂y/∂x = … ? ​