find the partial derivative.\nf(x,y)=6x + 2x²y² - 5y². find f_x(x,y)\na. 6 + 4xy²\nb. 4xy² - 10y\nc. 4x²y…
find the partial derivative.\nf(x,y)=6x + 2x²y² - 5y². find f_x(x,y)\na. 6 + 4xy²\nb. 4xy² - 10y\nc. 4x²y - 10y\nd. 6 + 4x²y
Answer
Explanation:
Step1: Differentiate each term with respect to (x)
- For the term (6x), using the power rule (\frac{d}{dx}(ax^n)=nax^{n - 1}), when (n = 1) and (a=6), (\frac{\partial(6x)}{\partial x}=6).
- For the term (2x^{2}y^{2}), treat (y) as a constant. Using the power rule (\frac{\partial(2x^{2}y^{2})}{\partial x}=2y^{2}\frac{\partial(x^{2})}{\partial x}). Since (\frac{\partial(x^{2})}{\partial x}=2x), then (2y^{2}\times2x = 4xy^{2}).
- For the term (-5y^{2}), since it does not contain (x), (\frac{\partial(-5y^{2})}{\partial x}=0).
Step2: Sum up the derivatives of each term
(f_{x}(x,y)=\frac{\partial(6x)}{\partial x}+\frac{\partial(2x^{2}y^{2})}{\partial x}+\frac{\partial(-5y^{2})}{\partial x}) (f_{x}(x,y)=6 + 4xy^{2}+0)
Answer:
A. (6 + 4xy^{2})