FA_LF_DiscussionWK15

Question 9 Ex12.4 Pg 711

Find \(f_x, f_y, f_{xx}, f_{yy}, f_{xy}, \:and\: f_{yx}\)

Solution

Function: \[ \begin{aligned} f(x, y) &= x^2y + 3x^2+4y-5 \end{aligned} \]

\(f_x\):

\[ \begin{aligned} f_x = 2xy + 6x \end{aligned} \]

Using R

library(calculus)

# define function
f <- expression((x**2)*y + 3*x**2 + 4*y - 5)

# partial derivative in respect to x
fx <- D(f, 'x')
print(fx)

## 2 * x * y + 3 * (2 * x)

\(f_y\):

\[ \begin{aligned} f_y = x^2 + 4 \end{aligned} \]

Using R

fy <- D(f, 'y')
print(fy)

## (x^2) + 4

\(f_{xx}\):

\[ \begin{aligned} f_{xx} = 2y + 6 \end{aligned} \]

Using R

# partial derivative in respect to x
fxx <- D(fx, 'x')
print(fxx)

## 2 * y + 3 * 2

\(f_{yy}\):

\[ \begin{aligned} f_{yy} = 0 \end{aligned} \]

Using R

fyy <- D(fy, 'y')
print(fyy)

## [1] 0

\(f_{xy}\):

\[ \begin{aligned} f_{xy} = 2x \end{aligned} \]

Using R

fxy <- D(fx, 'y')
print(fxy)

## 2 * x

\(f_{yx}\):

\[ \begin{aligned} f_{yx} = 2x \end{aligned} \]

Using R

fyx <- D(fy, 'x')
print(fyx)

## 2 * x