goaway716 said:
Find second order partial derivative of the function:
f(x,y)=x²y²+y²e^(x+y)
check that f_xy=f_yx
When differentiating w.r.t x, hold y as a constant.
\(\displaystyle x^{2}(0)+y^{2}\cdot 2x+y^{2}e^{x+y}+e^{x+y}(0)=2xy^{2}+y^{2}e^{x+y}\)
Now, do it again for the second partial.
\(\displaystyle 2x(0)+2y^{2}+y^{2}e^{x+y}+e^{x+y}(0)=2y^{2}+y^{2}e^{x+y}\)
Note, I used the product rule. If it helps to envision, let y be a constant such as 1. Remember, the derivative of a constant is 0.
Now, for the one above, try it w.r.t y letting x be the constant.
For \(\displaystyle f_{xy}\), first differentiate w.r.t x and then y.
For \(\displaystyle f_{yx}\), first differentiate w.r.t y and then x.
