calculate ă/Äx and ă/Äy. ƒ(x,y) = xy² + x²y
I don't know what your Ä term is supposed to represent. If it is supposed to be the "lazy-d" "del" ∂ symbol of a partial derivative, then
∂f/∂x = y^2 + 2xy, and
∂f/∂y = 2yx + x^2
To calculate the partial derivatives (∂ƒ/∂x) and (∂ƒ/∂y) of the function ƒ(x, y) = xy² + x²y, we need to find the derivative of ƒ with respect to x (∂ƒ/∂x) and with respect to y (∂ƒ/∂y) separately.
First, let's find (∂ƒ/∂x):
To calculate (∂ƒ/∂x), we differentiate ƒ(x, y) with respect to x while treating y as a constant.
Differentiating the first term, xy², with respect to x, we get y². And differentiating the second term, x²y, with respect to x, we get 2xy.
Therefore, (∂ƒ/∂x) = y² + 2xy.
Next, let's find (∂ƒ/∂y):
To calculate (∂ƒ/∂y), we differentiate ƒ(x, y) with respect to y while treating x as a constant.
Differentiating the first term, xy², with respect to y, we get 2xy. And differentiating the second term, x²y, with respect to y, we get x².
Therefore, (∂ƒ/∂y) = 2xy + x².
So, (∂ƒ/∂x) = y² + 2xy and (∂ƒ/∂y) = 2xy + x².