I've been asked to find the partial derivative with respect to x for the function:

$\displaystyle

f(x,y) = 2xe^{x+y^2} - e^y + sinh(2x + y)

$

I've learnt about treating y as though it's a constant, and this is what I've got so far. However, I think my working is wrong or my rules haven't been applied correctly. My logic was to apply the product rule to the first grouping, treat e^y just as a constant and just differentiate the last grouping based on the rules for sinh. Here's the result I got:

$\displaystyle

f_x(x,y) = 2x({e^{x+y^2}}\cdot{1}) + {e^{x+y^2}}\cdot{2} - 0 + {cosh(2x + y)}\cdot{2}

$

$\displaystyle

f_x(x,y) = 2e^{x+y^2}(x + 1) + 2cosh(2x+y)

$

Hopefully someone can shed some light on whether the working/answer is correct? I'm not feeling very confident about it.