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$Use the chain rule to find the indicated partial derivative {:[z=x^(4)+x^(2)y","quad x=s+2t-u","quad y=st^(2)],[(del z)/(del s)","(del z)/(del t)","(del z)/(del u)" when "s=3","t=1","u=2]:} (del z)/(del s)=$

Use the chain rule to find the indicated partial derivative $\newline$ $\begin{array}{l} z=x^{4}+x^{2} y, \quad x=s+2 t-u, \quad y=s t^{2} \\ \frac{\partial z}{\partial s}, \frac{\partial z}{\partial t}, \frac{\partial z}{\partial u} \text { when } s=3, t=1, u=2 \end{array}$ $\newline$ $\frac{\partial z}{\partial s}=$

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Calculus

Find higher derivatives of rational and radical functions

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Q. Use the chain rule to find the indicated partial derivative

\newline

\begin{array}{l} z=x^{4}+x^{2} y, \quad x=s+2 t-u, \quad y=s t^{2} \\ \frac{\partial z}{\partial s}, \frac{\partial z}{\partial t}, \frac{\partial z}{\partial u} \text { when } s=3, t=1, u=2 \end{array}

\newline

\frac{\partial z}{\partial s}=

Find Partial Derivative: First, we need to find the partial derivative of $z$ with respect to $s$ , using the chain rule. The chain rule states that if a variable $z$ depends on two other variables $x$ and $y$ , which in turn depend on a third variable $s$ , then the partial derivative of $z$ with respect to $s$ is given by: $\newline$ $\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\cdot\frac{\partial y}{\partial s}$
Calculate Derivatives: We need to calculate the partial derivatives that appear in the chain rule formula. First, let's find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ : $\newline$ $z = x^4 + x^2y$ $\newline$ $\frac{\partial z}{\partial x} = 4x^3 + 2xy$ $\newline$ $\frac{\partial z}{\partial y} = x^2$
Substitute Values: Next, we find the partial derivatives of $x$ and $y$ with respect to $s$ :
$x = s + 2t - u$
$\frac{\partial x}{\partial s} = 1$
$y = st^2$
$\frac{\partial y}{\partial s} = t^2$
Evaluate Expression: Now we substitute the values of $(\frac{\partial z}{\partial x}), (\frac{\partial z}{\partial y}), (\frac{\partial x}{\partial s}),$ and $(\frac{\partial y}{\partial s})$ into the chain rule formula: $\newline$ $(\frac{\partial z}{\partial s}) = (4x^3 + 2xy)(1) + x^2(t^2)$
Find Values: We need to evaluate this expression at the given values of $s$ , $t$ , and $u$ . First, we find the values of $x$ and $y$ at $s=3$ , $t=1$ , $u=2$ :
$x = s + 2t - u = 3 + 2\cdot1 - 2 = 3$
$y = st^2 = 3\cdot1^2 = 3$
Substitute Values: Now we substitute the values of $x$ and $y$ into the expression for $\frac{\delta z}{\delta s}$ : $\frac{\delta z}{\delta s} = (4\cdot3^3 + 2\cdot3\cdot3)\cdot(1) + (3^2)\cdot(1^2)$
Calculate Result: Finally, we calculate the value of $(\frac{\partial z}{\partial s}):$ $(\frac{\partial z}{\partial s}) = (4\cdot 27 + 2\cdot 9) + 9$ $(\frac{\partial z}{\partial s}) = (108 + 18) + 9$ $(\frac{\partial z}{\partial s}) = 126 + 9$ $(\frac{\partial z}{\partial s}) = 135$