Question

Find all first and second order partial derivatives for each of the following functions.

(Find fx(x, y), fy(x, y), fxx(x, y), fyy(x, y), fyx(x, y) and fxy(x, y).)

z = f(x, y) = ln (x^2 + 2xy + y^2 )

Answer 1

(a)

The function is .

Apply partial derivative on each side with respect to .

.

Answer 2

(b)

The function is .

Apply partial derivative on each side with respect to .

.

Answer 3

(c)

The function is .

Find .

From part (a): .

Apply partial derivative on each side with respect to .

Power rule of partial derivatives: .

.

Answer 4

(d)

The function is .

Find .

From part (b): .

Apply partial derivative on each side with respect to .

Power rule of partial derivatives: .

.

Answer 5

(e)

The function is .

Find .

From part (b): .

Apply partial derivative on each side with respect to .

.

Answer 6

(f)

The function is .

Find .

From part (a): .

Apply partial derivative on each side with respect to .