PROBLEMS ON PARTIAL DERIVATES IN THREE VARAIABLES

Problem 1 :

If

U(x, y, z) = (x²+y²)/xy + 3z²y

find (∂u/∂x), (∂u/∂y) and (∂u/∂z)

Solution :

U(x, y, z) = (x²+y²)/xy + 3z²y

Using quotient rule, we get

Differentiate with respect to x, so keep y and z as constants. 

∂u/∂x = [xy(2x)-(x²+y²)y]/(xy)² + 0

∂u/∂x = [2x²y-x²y+y³]/(xy)²

∂u/∂x = (x²y+y³)/(xy)²

∂u/∂x = (x²+y²)/(x²y)

Differentiate with respect to y, so keep x and z as constants. 

∂u/∂y = [xy(2y)-(x²+y²)x]/(xy)²+3z²(1)

∂u/∂y = [2xy²-x³-xy²]/(xy)²+3z²

∂u/∂y = (xy²-x³)/(xy)²+3z²

∂u/∂y = (y²-x²)/(xy²)+3z²

Differentiate with respect to z, so keep x and y as constants. 

∂u/∂z = 0+3(2z)y

∂u/∂z = 6zy

Problem 2 :

If U(x, y, z) = log (x³+y³+z³), find 

(∂u/∂x) + (∂u/∂y) + (∂u/∂z)

Solution :

U(x, y, z) = log (x³+y³+z³)

Finding ∂u/∂x :

Differentiate with respect to x. Treat y and z as constant.

∂u/∂x = [1/(x³+y³+z³)] (3x²)

∂u/∂x = [3x²/(x³+y³+z³)]  ------(1)

∂u/∂y = [1/(x³+y³+z³)] (3y²)

∂u/∂y = [3y²/(x³+y³+z³)]  ------(2)

∂u/∂z = [1/(x³+y³+z³)] (3z²)

∂u/∂y = [3z²/(x³+y³+z³)]  ------(3)

(1) + (2) + (3)

(∂u/∂x) + (∂u/∂y) + (∂u/∂z) 

= [3x²/(x³+y³+z³)] + [3y²/(x³+y³+z³)] + [3z²/(x³+y³+z³)]

=  3(x²+y²+z²)/(x³+y³+z³)

Problem 3 :

For each of the following functions find g_xy, g_xx, g_yy,and g_yx.

(i)  g(x, y) = xe^y+3x²y

Solution :

Finding g_x :

Differentiating respect to x.

g_x = e^y+3(2x)y

g_x = e^y+6xy

Finding g_y :

Differentiating respect to y.

g_y = xe^y+3x²(1)

g_y = xe^y+3x²

(i) Finding g_xy :

g_xy = e^y+6x(1)

g_xy = e^y+6x

(ii) Finding g_xx :

Differentiating respect to x.

g_x = e^y+6xy

g_xx = 0+6(1)y

g_xx = 6y

(iii) Finding g_yy :

g_y = xe^y+3x²

g_yy = xe^y+0

g_yy = xe^y

(iv) Finding g_yx :

g_y = xe^y+3x²

g_yx = (1)e^y+3(2x)

g_yx = e^y + 6x

(ii)  g(x, y) = log(5x+3y)

Solution :

Finding g_x :

g_x  =  [1/(5x+3y)] (5)

g_x  =  5/(5x+3y)

Finding g_y :

g_y  =  [1/(5x+3y)] (3)

g_y  =  3/(5x+3y)

(i) Finding g_xy :

From g_x  =  5/(5x+3y)

Differentiate with respect to y.

g_xy = [-5/(5x+3y)²](3)

g_xy = [-15/(5x+3y)²]

(ii) Finding g_xx :

From g_x  =  5/(5x+3y)

Differentiating respect to x.

g_xx =  [-5/(5x+3y)²](5)

g_xx = [-25/(5x+3y)²]

(iii) Finding g_yy :

From g_y = 3/(5x+3y)

Differentiating respect to y.

g_yy = -9/(5x+3y)²

(iv) Finding g_yx :

From g_y  =  3/(5x+3y)

Differentiating respect to x.

g_yx = -15/(5x+3y)²

(iii) g(x, y) = x²+3xy-7y+cos(5x)

Solution :

Finding g_x :

g_x = 2x+3y-0-sinx(5x)(5)

g_x = 2x+3y-5sinx(5x)

Finding g_y :

g_y = 0+3x(1)-7+0

g_y = 3x-7

(i) Finding g_xy :

From g_x = 2x+3y-5sinx(5x)

Differentiate with respect to y.

g_xy = 0+3+0

g_xy = 3

(ii) Finding g_xx :

From g_x = 2x+3y-5sinx(5x)

Differentiate with respect to x.

g_xx = 2+0-5cos(5x)(5)

g_xx = 2-25cos(5x)

(iii) Finding g_yy :

From g_y = 3x-7

Differentiate with respect to y.

g_yy = 0

(iv) Finding g_yx :

From g_y = 3x-7

Differentiate with respect to x.

g_yy = 3

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