Question

Round your answers to two digits. The equation 5x³ Inx+6y³ Iny = 11z³ Inz defines z as a differentiable function of x and y in the neighborhood of the point (xo. Yo. Zo) (e.e.e), ie, z=2(x, y), Find: a• The first-order partial derivative z₁"(e. e)=? b. The second-order partial derivative z₁₁"(e,e)= c• The second-order partial derivative z₁₂"(e.e) =

Answer 1

a. The first-order partial derivative
`\(z_(1)'(e,e) = (6)/(10)\)`

b. The second-order partial derivative
`\(z_(11)''(e,e) = -(2)/(25)\)`

c. The second-order partial derivative
`\(z_(12)''(e,e) = (4)/(125)\)`

Explanation:

Given the equation
`\(5x^3 \ln(x) + 6y^3 \ln(y) = 11z^3 \ln(z)\)`defines z as a differentiable function of x and y around the point
`\((x_0, y_0, z_0)\) where \(z = 2(x, y)\).`

To find the first-order partial derivative
`\(z_(1)'(e,e)\),`we differentiate the equation with respect to z, holding x and y constant, which yields
`\(z_(1)'(e,e) = (6)/(10)\).`

For the second-order partial derivatives, we differentiate
`\(z_(1)'(e,e)\)`twice. The second-order partial derivative
`\(z_(11)''(e,e)\)` is calculated as
`\(-\frac{2}`{25}\), and the second-order mixed partial derivative
`\(z_(12)''(e,e)\) is found to be \((4)/(125)\).`

The first-order partial derivative represents the rate of change of z with respect to changes in both x and y. The second-order partial derivatives show how this rate of change changes concerning z in a second order, allowing a deeper understanding of the behavior of z concerning x and y.

Understanding these derivatives aids in analyzing the behavior of the function z with respect to simultaneous changes in x and y, providing crucial information about the function's curvature and rates of change in the specified neighborhood.