Question

find the partial derivative.
f(x,y)=6x + 2x²y² - 5y². find f_x(x,y)
a. 6 + 4xy²
b. 4xy² - 10y
c. 4x²y - 10y
d. 6 + 4x²y

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. \(6 + 4xy^{2}\)