find the partial derivative.\nf(x,y)=3x + 3x^{2}y^{2}-8y^{2}. find f_{x}(x,y)\na. 3 + 6xy^{2}\nb…

Explanation:

Step1: Differentiate each term with respect to (x)

• Differentiate (3x) with respect to (x): The derivative of (ax^n) with (a = 3) and (n=1) is (a\times n\times x^{n - 1}). So, (\frac{\partial(3x)}{\partial x}=3).
• Differentiate (3x^{2}y^{2}) with respect to (x): Using the power - rule (\frac{\partial(ax^{n}y^{m})}{\partial x}=a\times n\times x^{n - 1}y^{m}) (treating (y) as a constant). Here (a = 3), (n = 2), (m=2), so (\frac{\partial(3x^{2}y^{2})}{\partial x}=3\times2\times x^{2-1}y^{2}=6xy^{2}).
• Differentiate (-8y^{2}) with respect to (x): Since (y) is treated as a constant, (\frac{\partial(-8y^{2})}{\partial x}=0).

Step2: Sum up the derivatives

(f_{x}(x,y)=\frac{\partial(3x)}{\partial x}+\frac{\partial(3x^{2}y^{2})}{\partial x}+\frac{\partial(-8y^{2})}{\partial x}) (f_{x}(x,y)=3 + 6xy^{2}+0)

Answer:

A. (3 + 6xy^{2})