Given the function $f(x,y)=ax+by+cx+dy$, where $a$, $b$, $c$, and $d$ are constants, find the first partial derivatives $f_x$ and $f_y$.

• A. $f_x=a+c$, $f_y=b+d$
• B. $f_x=c$, $f_y=d$
• C. $f_x=a$, $f_y=b$
• D. $f_x=ax+cx$, $f_y=by+dy$

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the concept of partial derivatives. A partial derivative measures how a function changes as one of its variables changes, while keeping the other variables constant.
2. Step 2: Identify the function f(x, y) = a x + b y + c x + d y. Note that the function is linear in both x and y, and the constants a, b, c, and d are coefficients.
3. Step 3: To find the partial derivative with respect to x (denoted as f_x), treat y as a constant and differentiate the terms involving x. The derivative of ax with respect to x is a, and the derivative of cx with respect to x is c. Combine these results.
4. Step 4: To find the partial derivative with respect to y (denoted as f_y), treat x as a constant and differentiate the terms involving y. The derivative of by with respect to y is b, and the derivative of dy with respect to y is d. Combine these results.
5. Step 5: Write the final expressions for the partial derivatives: f_x = a + c and f_y = b + d. These represent the rates of change of the function with respect to x and y, respectively.