Question

what are the first order partial derivatives of f(x,y) = x*e^y

Answer 1

Well, if you want to differentiate with respect to x, you treat anything that is not x as a constant term. So looking at xe^y, you can think of e^y as a mere coefficient. Well, when you take a derivative, coefficients just stay put. So for partial with respect to x, think of it as differentiating only x and then having it multiplied by the constant e^y. Which means
\[f_{x} = (1)*e^{y} = e^{y}\]

The same thing applies for the partial with respect to y. Now we think of everything that is not y as a constant. So it's like taking the derivative of e^y and then just multiplying by the coefficient that is just sitting there. So that means for the partial with respect to y we have:
\[f_{y} = \frac{ \delta }{ \delta y }[xe^{y}] = xe^{y}\]. the derivative of e^y is just e^y and any multiplicative constant just stays put, so the derivative is itself. Hope that makes sense somewhat :)

Answer 2

oh awesome! Its just sometimes I forget what is and isnt acting as a constant. Thanks for your help :)

Answer 3

If it helps, you can try and move everything that is being kept constant to the front, that way you can ignore it. So for example, if I needed the partial derivative with respect to y of
\[f(x,y,z) = \sqrt{x}*y^{3}\ln(z)\], I might just rewrite it like this:
\[[\sqrt{x}*\ln(z)]*y^{3}\]It's a way of highlighting what is and is not constant. From there, the derivative of the y portion is simply 3y^2, which is then multiplied by sqrt(x)*ln(z) to get your partial derivative :)

Answer 4

great! that's actually really helpful.

Answer 5

Sure, glad to help :) Good luck!