Question

Partial derivatives. For each of the following multi-variable functions, compute the partial derivatives with respect to both x and y.

a. f(x,y) = xy
b. f(x,y) = 2x + 3y
c. f(x, y) = x²y
d. f(x,y) = ln(x) + 2 ln(y)
e. f(x, y) = 2√x + 2√y
f. f(x,y) = x/y
g. f(x,y) = y/х

Answer 1

Partial derivatives of a function are found by differentiating with respect to one variable while keeping the other constant. For each given multi-variable function, the partial derivatives with respect to x and y are calculated and presented.

Step-by-step explanation:

To find the partial derivatives of each multi-variable function respect to both x and y, we keep one variable constant while differentiating with respect to the other.

• f(x, y) = xy:
  Partial derivative with respect to x: f_x = y
  Partial derivative with respect to y: f_y = x
• f(x, y) = 2x + 3y:
  Partial derivative with respect to x: f_x = 2
  Partial derivative with respect to y: f_y = 3
• f(x,y) = x²y:
  Partial derivative with respect to x: f_x = 2xy
  Partial derivative with respect to y: f_y = x²
• f(x,y) = ln(x) + 2 ln(y):
  Partial derivative with respect to x: f_x = 1/x
  Partial derivative with respect to y: f_y = 2/y
• f(x, y) = 2√x + 2√y:
  Partial derivative with respect to x: f_x = 1/√x
  Partial derivative with respect to y: f_y = 1/√y
• f(x,y) = x/y:
  Partial derivative with respect to x: f_x = 1/y
  Partial derivative with respect to y: f_y = -x/y²
• f(x,y) = y/x:
  Partial derivative with respect to x: f_x = -y/x²
  Partial derivative with respect to y: f_y = 1/x