Partial derivative with respect to y of g(x,y) = x*e^y?

Question

I'm unsure of whether this derivative is equal to x*e^y, or x*e^y + e^y.
Is this equation different from the deriving the equation x = x*e^y => dx/dy = x*e^y? 
Is x treated as a constant in this situation? I think I'm just generally confused on the difference between partial derivatives and regular derivatives. Can someone please explain?

xyzzy · Accepted Answer

if you take the partial derivative with respect to y then you will treat x as if it were a constant.

dg/dy = x e^y

sorry, I don't know the html to make the curly d's that identify them as partial deriv's

partial derivatives... if you think of g(x,y) then the partial derivative with respect to x would be a look at the slop of the paths made by a series of slices parallel to the x axis.  And dg/dy would the the paths parallel to the y axis.

you can link the partial deriv to the regular deriv as follows.

regular deriv...
dg/dt = dg/dx dx/dt  + dg/dy dy/dt

dg/dt is a regular deriv'
dg/dx and dg/dy are partial deriv's and dx/dt and dy/dt are regular deriv's.

if you have a parametric curve for x and y as a function of t, then dg/dt rate of change encountered as you follow the path of the contour.

or

dg/dx (regular) = dg/dx (partial) + dg/dy (partial) dy/dx (regular)

and if you play with this you will see that this is identical to "implicit differentiation" from calculus I.

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Partial derivative with respect to y of g(x,y) = x*e^y?

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