find all the critical points of the function
f(x,y) = (x-2y)e^(x^2-y^2)
and classify them as local maxima, local minima or saddle points.
How is this differentiated?
You need to find where $\displaystyle \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial x}\right) = 0 $
To find $\displaystyle \frac{\partial f}{\partial x}$ differentiate $\displaystyle f(x,y)$ with respect to x holding y constant
To find $\displaystyle \frac{\partial f}{\partial y}$ differentiate $\displaystyle f(x,y)$ with respect to y holding x constant
You need to use the product rule it is... $\displaystyle f = uv \Rightarrow f'=uv'+vu'$
Differentiating with respect to x (finding $\displaystyle \frac{\partial f}{\partial x}$ ) make
$\displaystyle u = x-2y \Rightarrow u' = 1$
$\displaystyle v = e^{x^2-y^2} \Rightarrow v' = 2x\times e^{x^2-y^2}$
Now apply $\displaystyle f = uv \Rightarrow f'=uv'+vu'$
After this you need to differentiate for y. (finding $\displaystyle \frac{\partial f}{\partial y}$ )
Then solve for $\displaystyle \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial x}\right) = 0$