Question : Find the second-order partial derivative. 19) Let H(x, y) = (3xy/x^2 - y). Find : 2163353

Find the second-order partial derivative.

19) Let H(x, y) = (3xy/x² - y). Find (∂²H/∂y²).

A) (9x⁴ - 6x³ - 9x²y/(x² - y)³)

B) (3x³/(x² - y)²)

C) (6x³/(x² - y)³)

D) (6x³y - 6x⁵/(x² - y)⁴)

E) none of these

20) Let F(x, y, z) = (xz/y²z + x) - 5x²y³. Compute (∂F/∂y)(-1, 0, 1).

A) - (1/2)

B) 2

C) (3/5)

D) 0

E) none of these

21) Let P(x, y, z) = xy + 2x³√(y² - 1). Compute (∂P/∂z)(2, 1, 3).

A) 1

B) 3

C) 2

D) 0

E) none of these

22) Let f(x, y) = x³y + e^{x + 3y}. Compute (∂²f/∂x ∂y)(1, 0).

A) 3 + e

B) 3 + 3e

C) 6 + e

D) 6

E) none of these

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

23) Let f(x, y) = (x² + x)e^{y - x}. Find (∂f/∂x).

Enter your answer as just (P(x))e^{y - x} where P is a polynomial in x in standard form.

24) Let f(x, y) = 4y² - 2x³ + 5xy². Find (∂f/∂x).

Enter just a polynomial in x plus or minus a polynomial in y both in standard form (do not label, no parentheses).

25) Let f(x, y) = e^{x^2y}. Find (∂f/∂y).

Enter your answer exactly as just e^{x^2y}(P(x)) where P is a polynomial in x in standard form (do not label).

26) Let f(x, y) = x² + y. Find (∂f/∂y).

Enter just an integer.

27) Let f(x, y) = 3x² + 2xy. Find (∂f/∂x).

Enter your answer exactly as just a polynomial in x plus or minus a polynomial in y both in standard form (do not label, no parentheses).

28) Let f(x, y) = x² + 2xy + e^y. Find (∂f/∂y).

Enter just a polynomial in x plus or minus a polynomial in e^y both in standard form (do not label, no parentheses).

29) Let f(x, y) = 4x² + 2y^2/3 + e^{x^4/3}. Find (∂f/∂y).

Enter just a power function in y in standard form (do not label).

30) Let f(x, y) = 4x² + 2y² + 3xy. Find (∂f/∂y).

Enter just a polynomial in y plus or minus a polynomial in x both in standard form (no label, no parentheses).

31) Let f(x, y) = 2y³ + 4x⁴ + 2xy. Find (∂f/∂y).

Enter just a polynomial in y plus or minus a polynomial in x both in standard form (no label, no parentheses).

32) Let f(x, y, z) = ln(xy²z³). Find (∂f/∂z).

Enter your answer in the unlabeled form az^b.

33) Let f(x, y) = x²y + y²x + 2xy. Find (∂²f/∂x ∂y).

Enter just a polynomial in x plus or minus a polynomial in y plus or minus two, both polynomials in standard form (no label, no parentheses).

34) Let f(x, y) = (x/y + 1). Find (∂²f/∂x ∂y).

Enter just ±(P(y))^a where P is a polynomial in standard form (do not label).

35) Let f(x, y) = y(x + e^{y - x}). Compute (∂f/∂x)(2, 3).

Enter your answer exactly as just a(b ± E).

36) Let f(x, y) = xy. Find (∂f/∂x) and (∂f/∂y). Is y, x correct in the corresponding order?

Enter "yes" or "no".

37) Let f(x, y) = x²y - xy². Find (∂f/∂x). Is y² + 2xy the correct answer?

Enter "yes" or "no".