Question

use the chain rule to find $\frac{partial z}{partial s}$ and $\frac{partial z}{partial t}$, where $z = x^{2}+xy + y^{2}$, $x = 9s+6t$, $y = 7s + 10t$. first the pieces: $\frac{partial z}{partial x}=$ $\frac{partial z}{partial y}=$ $\frac{partial x}{partial s}=$ $\frac{partial x}{partial t}=$ $\frac{partial y}{partial s}=$ $\frac{partial y}{partial t}=$ and putting it all together: $\frac{partial z}{partial s}=$ (in terms of $s$ and $t$) $\frac{partial z}{partial t}=$ (in terms of $s$ and $t$) submit answer next item

Explanation:

Step1: Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$

Given $z = x^{2}+xy + y^{2}$, then $\frac{\partial z}{\partial x}=2x + y$ and $\frac{\partial z}{\partial y}=x + 2y$.

Step2: Find $\frac{\partial x}{\partial s}$, $\frac{\partial x}{\partial t}$, $\frac{\partial y}{\partial s}$ and $\frac{\partial y}{\partial t}$

Since $x = 9s+6t$, then $\frac{\partial x}{\partial s}=9$ and $\frac{\partial x}{\partial t}=6$. Since $y = 7s + 10t$, then $\frac{\partial y}{\partial s}=7$ and $\frac{\partial y}{\partial t}=10$.

Step3: Use the chain - rule to find $\frac{\partial z}{\partial s}$

The chain - rule states that $\frac{\partial z}{\partial s}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$. Substitute the values we found: $\frac{\partial z}{\partial s}=(2x + y)\times9+(x + 2y)\times7$. Expand this: $\frac{\partial z}{\partial s}=18x+9y + 7x+14y=25x + 23y$. Then substitute $x = 9s+6t$ and $y = 7s + 10t$ into it: $\frac{\partial z}{\partial s}=25(9s + 6t)+23(7s + 10t)=225s+150t+161s + 230t=386s+380t$.

Step4: Use the chain - rule to find $\frac{\partial z}{\partial t}$

The chain - rule states that $\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$. Substitute the values we found: $\frac{\partial z}{\partial t}=(2x + y)\times6+(x + 2y)\times10$. Expand this: $\frac{\partial z}{\partial t}=12x+6y+10x + 20y=22x+26y$. Then substitute $x = 9s+6t$ and $y = 7s + 10t$ into it: $\frac{\partial z}{\partial t}=22(9s + 6t)+26(7s + 10t)=198s+132t+182s+260t=380s+392t$.

Answer:

$\frac{\partial z}{\partial x}=2x + y$, $\frac{\partial z}{\partial y}=x + 2y$, $\frac{\partial x}{\partial s}=9$, $\frac{\partial x}{\partial t}=6$, $\frac{\partial y}{\partial s}=7$, $\frac{\partial y}{\partial t}=10$, $\frac{\partial z}{\partial s}=386s + 380t$, $\frac{\partial z}{\partial t}=380s+392t$