Problem 38

Question

Evaluate $w_{x}, w_{y},$ and $w_{z}$ at the point. $$ \text { Function } \quad \text { Point } $$ $$ w=x y e^{z^{2}} \quad(2,1,0) $$

Step-by-Step Solution

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Answer

The evaluations of the partial derivatives $w_{x}$, $w_{y}$, and $w_{z}$ at the point (2,1,0) are: $w_{x} = 1$, $w_{y} = 2$, and $w_{z} = 0$.

1Step 1: Calculate the partial derivatives

The first step is to calculate the partial derivative with respect to each of the variables ($x, y, z$):For $w_{x} = \frac{\partial w}{\partial x}$,Keep $y$ and $z$ constant and differentiate $w=x y e^{z^{2}}$ with respect to $x$. It becomes: $w_{x} = y e^{z^{2}}$ For $w_{y} = \frac{\partial w}{\partial y}$,Keep $x$ and $z$ constant and differentiate $w= x y e^{z^{2}}$ with respect to $y$. It becomes: $w_{y} = x e^{z^{2}}$ For $w_{z} = \frac{\partial w}{\partial z}$,Keep $x$ and $y$ constant and differentiate $w= x y e^{z^{2}}$ with respect to $ z $. Here, we need to use the chain rule. The chain rule states that the derivative of $ f(g(x)) $ is $ f'(g(x)) \cdot g'(x) $. We consider $e^{z^{2}}$ as $f(z)$ and $z^{2}$ as $g(z)$. So, $f'(z)= e^{z^{2}}$, $g'(z)= 2z$,Then it becomes: $w_{z} = 2 x y z e^{z^{2}}$

2Step 2: Substitute the known values

We then substitute the values of $x=2, y=1, z=0$ into the expressions we obtained in step one:For $w_{x}$,Substituting $y=1, z=0$ into the equation, we get: $w_{x} = 1 e^{0} = 1$For $w_{y}$,Substituting $x=2, z=0$ into the equation, we get: $w_{y} = 2 e^{0} = 2$For $w_{z}$,Substituting $x=2, y=1, z=0$ into the equation, we get: $w_{z} = 2*2*1*0 e^{0} = 0$

3Step 3: Writing down the results

After substituting the values into the partial derivative expressions, the resulting values for the partial derivatives $w_{x}$, $w_{y}$, and $w_{z}$ at the point (2,1,0) are: $w_{x} = 1$, $w_{y} = 2$, and $w_{z} = 0$. These are derived as the rate of change of the function $w$ w.r.t $x$ is 1, w.r.t $y$ is 2 and w.r.t $z$ is 0 at the point (2,1,0).

Key Concepts

Chain RuleRate of ChangePartial Derivative Calculations

Chain Rule

The Chain Rule is a fundamental concept in calculus, particularly useful when dealing with composite functions. It allows us to differentiate functions that are nested within each other, like in our case with the function $ w = xy e^{z^2} $. Here, $ e^{z^2} $ is a nested component, making the Chain Rule indispensable for partial derivative calculations.
When we apply the chain rule, we find the derivative of the outer function and multiply it by the derivative of the inner function. For $ w_z $, we consider $ f(z) = e^{z^2} $ as the outer function and $ g(z) = z^2 $ as the inner function. Thus,

The derivative of the outer function $ f'(z) $ is $ e^{z^2} $.
The derivative of the inner function $ g'(z) $ is $ 2z $.

Combining these with the Chain Rule, $ w_z = \frac{\partial w}{\partial z} = 2xyz e^{z^2} $. It's crucial to understand this process as it enables us to differentiate more complex expressions efficiently.

Rate of Change

Rate of Change is a core idea in calculus used to describe how a quantity changes with respect to another variable. In the context of partial derivatives, it tracks the local behavior of a function as individual variables change.
In our exercise, we're focused on the partial derivatives of $ w = xy e^{z^2} $, which means examining how $ w $ changes when we tweak $ x $, $ y $, and $ z $ independently. Each partial derivative we compute tells us something about the sensitivity of $ w $:

$ w_x = y e^{z^2} $ indicates how $ w $ changes as $ x $ changes while keeping $ y $ and $ z $ constant.
$ w_y = x e^{z^2} $ shows the effect of changing $ y $ alone.
$ w_z = 2xyz e^{z^2} $ describes how quickly $ w $ changes with shifts in $ z $, thanks to the Chain Rule.

These derivatives at the point (2, 1, 0) reflect specific rates: $ w_x = 1 $, $ w_y = 2 $, and $ w_z = 0 $, highlighting the nuances in how each variable affects the function's output.

Partial Derivative Calculations

Partial Derivative Calculations are vital when dealing with functions of multiple variables. They allow us to isolate and understand the effect of each variable independently on the function's overall behavior.
In our example, the function $ w = xy e^{z^2} $ requires finding derivatives with respect to $ x $, $ y $, and $ z $. This involves holding the other variables constant and differentiating with respect to the one of interest. For each variable, we follow these steps:

For $ w_x $, treat $ y $ and $ z $ as constants, hence, $ w_x = y e^{z^2} $.
For $ w_y $, use the same principle with $ x $ and $ z $ held fixed: $ w_y = x e^{z^2} $.
$ w_x $, $ w_y $, and especially $ w_z $ use patterns like chain rule interactions for more complex terms.

Substituting the given point (2, 1, 0) into these derivatives, we can clearly see the isolated effects, simplifying the interpretation to: $ w_x = 1 $, $ w_y = 2 $, and $ w_z = 0 $. Mastery of these calculations helps in understanding and modeling systems in physics, engineering, and beyond easily.

Problem 38

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