Partial derivative of a single variable function

[wumple]

So I don't understand why if you have something like U(x,y) = f(y+2x)

and you take \frac{\partial U}{\partial x}<br /> = \frac{\partial f}{\partial x}

you get \frac{df}{d(y+2x)} * \frac{d(y+2x)}{dx}

Why does the partial derivative just change to the total derivative for one variable? It seems like you just treat x and y+2x as the same variable? thanks for any help!

 

[Fredrik]

Define a function g by g(x,y)=y+2x for all x,y. You have specified that the relationship between U, f and g is given by
$$U(x,y)=f(g(x,y))=(f\circ g)(x,y)$$ for all x,y. So $U=f\circ g$, and therefore $$\frac{\partial U}{\partial x}=\frac{\partial (f\circ g)}{\partial x}.$$ Since we're dealing with a composition of functions, we need the chain rule. It tells us that the right-hand side is $$=f'\, \frac{\partial g}{\partial x}.$$ This is often written as $$\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}.$$ The g in the denominator can serve as a reminder that f' is to be evaluated at g(x,y).

 

[wumple]

Thanks! I just wasn't thinking of y+2x as a separate function of x and y.