find the partial derivative.\nf(x,y)=8x + 4x^{2}y^{2}-3y^{2}. find f_{x}(x,y).\na. 8x^{2}y - 6y\nb. 8 +…
find the partial derivative.\nf(x,y)=8x + 4x^{2}y^{2}-3y^{2}. find f_{x}(x,y).\na. 8x^{2}y - 6y\nb. 8 + 8xy^{2}\nc. 8 + 8x^{2}y\nd. 8xy^{2}-6y
Answer
Explanation:
Step1: Differentiate each term with respect to (x)
- For the term (8x), using the power rule (\frac{d}{dx}(ax^n)=anx^{n - 1}) (here (n = 1)), (\frac{\partial(8x)}{\partial x}=8).
- For the term (4x^{2}y^{2}), using the power rule (\frac{\partial(4x^{2}y^{2})}{\partial x}=4y^{2}\frac{\partial(x^{2})}{\partial x}). Since (\frac{\partial(x^{2})}{\partial x}=2x), then (4y^{2}\times2x = 8xy^{2}).
- For the term (-3y^{2}), since (y) is treated as a constant when differentiating with respect to (x), (\frac{\partial(-3y^{2})}{\partial x}=0).
Step2: Sum up the derivatives of each term
(f_{x}(x,y)=\frac{\partial(8x)}{\partial x}+\frac{\partial(4x^{2}y^{2})}{\partial x}+\frac{\partial(-3y^{2})}{\partial x}=8 + 8xy^{2}+0)
Answer:
B. (8 + 8xy^{2})