Find f_x and f_y for the following multi-variable functions.

1. f(x,y) = xy²
2. f(x,y) = 2x³y² + y
3. f(x,y) = 1.45x²y - xy + 2x - 3
4. f(x,y) = (x/y)
5. f(x,y) = x^y
6. f(x,y) = x ln y
7. f(x,y) = 2e^xy
8. f(x,y) = 3^xy + x³y²
9. f(x,y) = 2x³(1+y²)²
10. f(x,y) = (1+x²y)³

Find all second derivatives (f_xx, f_xy, f_yx and f_yy) of these functions.

1. f(x,y) = 2x⁵y - x²
2. f(x,y) = x + 3y²
3. f(x,y) = xe^y
4. f(x,y) = ln(x² + y²)
5. f(x,y) = 2^y3^x

Answers

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1. f_x = y²; f_y = 2xy
2. f_x = 6x²y²; f_y = 4x³y + 1
3. f_x = 2.9xy - y + 2; f_y = 1.45x² - x
4. f_x = 1/y; f_y = -x/y²
5. f_x = yx^y-1; f_y = x^y(ln x)
6. f_x = ln y; f_y = x/y
7. f_x = 2ye^xy; f_y = 2xe^xy
8. f_x = y3^xy(ln 3) + 3x²y²; f_y = x3^xy(ln 3) + 2x³y
9. f_x = 6x²(1 + y²)²; f_y = 8x³y(1 + y²)
10. f_x = 6xy(1 + x²y)²; f_y = 3x²(1 + x²y)²

1. f_x = 10x⁴y - 2x; f_y = 2x⁵; f_xx = 40x³y - 2; f_xy = 10x⁴; f_yx = 10x⁴; f_yy = 0
2. f_x = 1; f_y = 6y; f_xx = 0; f_xy = 0; f_yx = 0; f_yy = 6
3. f_x = e^y; f_y = xe^y; f_xx = 0; f_xy = e^y; f_yx = e^y; f_yy = xe^y
4. f_x = 2x/(x² + y²); f_y = 2y/(x² + y²); f_xx = (2y² - 2x²)/(x² + y²)²; f_xy = -4xy/(x² + y²)²; f_yx = -4xy/(x² + y²)²; f_yy = (2x² - 2y²)/(x² + y²)²
5. f_x = 2^y3^x(ln 3); f_y = 2^y3^x(ln 2); f_xx = 2^y3^x(ln 3)²; f_xy = 2^y3^x(ln 3)(ln 2); f_yx = 2^y3^x(ln 3)(ln 2); f_yy = 2^y3^x(ln 2)²

Errors corrected 9/22/06 - sas