let ( z = f(x,y) = 12x^{2}-16xy + 9y^{2} ). find the following using the formal definition of the partial…

let ( z = f(x,y) = 12x^{2}-16xy + 9y^{2} ). find the following using the formal definition of the partial derivative.\na. ( \frac{partial z}{partial x} )\nb. ( \frac{partial z}{partial y} )\nc. ( \frac{partial f}{partial x}(2,1) )\nd. ( f_{y}(2,3) )

let ( z = f(x,y) = 12x^{2}-16xy + 9y^{2} ). find the following using the formal definition of the partial derivative.\na. ( \frac{partial z}{partial x} )\nb. ( \frac{partial z}{partial y} )\nc. ( \frac{partial f}{partial x}(2,1) )\nd. ( f_{y}(2,3) )

Answer

Explanation:

Step1: Find the partial derivative with respect to (x)

For (z = f(x,y)=12x^{2}-16xy + 9y^{2}), using the power rule (\frac{\partial}{\partial x}(ax^{n}y^{m})=nax^{n - 1}y^{m}) (where (m) is a constant). (\frac{\partial z}{\partial x}=\frac{\partial}{\partial x}(12x^{2})-\frac{\partial}{\partial x}(16xy)+\frac{\partial}{\partial x}(9y^{2})) (\frac{\partial z}{\partial x}=24x-16y+0 = 24x-16y)

Step2: Find the partial derivative with respect to (y)

Using the power rule (\frac{\partial}{\partial y}(ax^{n}y^{m})=max^{n}y^{m - 1}) (where (n) is a constant). (\frac{\partial z}{\partial y}=\frac{\partial}{\partial y}(12x^{2})-\frac{\partial}{\partial y}(16xy)+\frac{\partial}{\partial y}(9y^{2})) (\frac{\partial z}{\partial y}=0-16x + 18y=18y-16x)

Step3: Evaluate (\frac{\partial f}{\partial x}) at ((2,1))

Substitute (x = 2) and (y = 1) into (\frac{\partial z}{\partial x}=24x-16y) (\frac{\partial f}{\partial x}(2,1)=24\times2-16\times1=48 - 16=32)

Step4: Evaluate (f_y) at ((2,3))

Since (f_y=\frac{\partial z}{\partial y}=18y-16x), substitute (x = 2) and (y = 3) (f_y(2,3)=18\times3-16\times2=54 - 32 = 22)

Answer:

a. (24x-16y) b. (18y - 16x) c. (32) d. (22)