Problem 8 Find all first and second partia... [FREE SOLUTION]

Chapter 13: Problem 8

Find all first and second partial derivatives of \(z\) with respect to \(x\) and \(y\) if \(x^{2}+4 y^{2}+\) \(16 z^{2}-64=0\)

Short Answer

Expert verified

The first partial derivatives are \(\frac{\partial z}{\partial x} = -\frac{x}{16z}\) and \(\frac{\partial z}{\partial y} = -\frac{y}{4z}\). The second partial derivatives are \(\frac{\partial^2 z}{\partial x^2} = \frac{-1}{16z} - \frac{x^2}{256z^3}\), \(\frac{\partial^2 z}{\partial y^2} = \frac{-1}{4z} - \frac{y^2}{16z^3}\), and \(\frac{\partial^2 z}{\partial x \partial y} = \frac{x y}{64z^3}\).

Step by step solution

Identify the Implicit Function

Given the implicit function: \[ x^2 + 4y^2 + 16z^2 - 64 = 0 \]We need to find partial derivatives of \(z\) with respect to \(x\) and \(y\).

Differentiate Implicitly with Respect to x

Differentiate each term in the given implicit equation with respect to \(x\):\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(4y^2) + \frac{d}{dx}(16z^2) - \frac{d}{dx}(64) = 0 \]This results in:\[ 2x + 0 + 32z \frac{\partial z}{\partial x} = 0 \]Thus, solving for \(\frac{\partial z}{\partial x}\):\[ \frac{\partial z}{\partial x} = -\frac{x}{16z} \]

Differentiate Implicitly with Respect to y

Differentiate each term in the given implicit equation with respect to \(y\):\[ \frac{d}{dy}(x^2) + \frac{d}{dy}(4y^2) + \frac{d}{dy}(16z^2) - \frac{d}{dy}(64) = 0 \]This simplifies to:\[ 0 + 8y + 32z \frac{\partial z}{\partial y} = 0 \]And solving for \(\frac{\partial z}{\partial y}\):\[ \frac{\partial z}{\partial y} = -\frac{y}{4z} \]

Find the Second Partial Derivative with Respect to x

Differentiate \(\frac{\partial z}{\partial x} = -\frac{x}{16z}\) with respect to \(x\) again:\[ \frac{\partial^2 z}{\partial x^2} = \frac{-1}{16z} + \frac{x}{16z^2} \frac{\partial z}{\partial x} = \frac{-1}{16z} - \frac{x^2}{256z^3} \] Using the chain rule, where previously \(\frac{\partial z}{\partial x} = -\frac{x}{16z}\).

Second Partial Derivative with Respect to y

Differentiate \(\frac{\partial z}{\partial y} = -\frac{y}{4z}\) with respect to \(y\):\[ \frac{\partial^2 z}{\partial y^2} = \frac{-1}{4z} - \frac{y^2}{16z^3} \]

Mixed Second Partial Derivative \( \frac{\partial^2 z}{\partial x \partial y} \)

Find the mixed second partial derivative using previously found \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \):Differentiating \( \frac{\partial z}{\partial x} = -\frac{x}{16z} \) with respect to \(y\) results in:\[ \frac{\partial^2 z}{\partial x \partial y} = \frac{x y}{64z^3} \]

Mixed Second Partial Derivative \( \frac{\partial^2 z}{\partial y \partial x} \)

Similarly, differentiating \( \frac{\partial z}{\partial y} = -\frac{y}{4z} \) with respect to \(x\) results in:\[ \frac{\partial^2 z}{\partial y \partial x} = \frac{x y}{64z^3} \] Since partial derivatives commute, the mixed derivatives are equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function that is not solely defined in terms of one variable. To use implicit differentiation, we take the derivative of each term with respect to a particular variable, noticing the relationships among all the involved variables.

In our exercise, we had the implicit equation: \[ x^2 + 4y^2 + 16z^2 - 64 = 0 \]
Instead of solving for \(z\) explicitly and then differentiating, which can be complex or impossible, we differentiated directly using variables \(x\), \(y\), and \(z\).
While differentiating terms involving \(z\), we multiply by the appropriate partial derivative, since \(z\) is essentially a function of both \(x\) and \(y\).

Through implicit differentiation, we found the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\).These show change rates of \(z\) as \(x\) and \(y\) vary, which are vital for understanding many multivariable functions.

Second Partial Derivative

Second partial derivatives provide deeper insights into how a function's curvature behaves. They are essentially the derivatives of the first partial derivatives. In our exercise, we calculated the second partial derivatives such as \(\frac{\partial^2 z}{\partial x^2}\) and \(\frac{\partial^2 z}{\partial y^2}\).

Let's consider the second partial derivative \(\frac{\partial^2 z}{\partial x^2}\), which involves differentiating \(\frac{\partial z}{\partial x}\) with respect to \(x\) again.
This results in \[ \frac{\partial^2 z}{\partial x^2} = \frac{-1}{16z} - \frac{x^2}{256z^3} \]showing how \(z\)'s slope with respect to \(x\) changes as \(x\) itself changes, further describing how the surface bends in the \(x\)-direction.
Similarly, we find the expression \(\frac{\partial^2 z}{\partial y^2}\) by applying the same method on \(\frac{\partial z}{\partial y}\).

These derivatives help analyze rates of change and curvature in specific directions, which are fundamental for applications like optimization, and understanding the geometry of multivariable graphs.

Mixed Partial Derivatives

Mixed partial derivatives involve taking derivatives with respect to two different variables consecutively. In the discussed problem, this would mean finding \(\frac{\partial^2 z}{\partial x \partial y}\) and \(\frac{\partial^2 z}{\partial y \partial x}\).

For \(\frac{\partial^2 z}{\partial x \partial y}\), we differentiate \(\frac{\partial z}{\partial x}\) with respect to \(y\).
The result is \[ \frac{\partial^2 z}{\partial x \partial y} = \frac{x y}{64z^3} \]emphasizing how the slope of \(z\) in the \(x\)-direction changes as \(y\) changes.
Due to the symmetry of mixed derivatives, known as Clairaut's Theorem, if the second mixed derivatives are continuous, \(\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x}\).

Mixed partial derivatives are crucial in modeling real-world phenomena where multiple variables interact, and assessing if these interactions have a symmetrical effect, as highlighted in this problem.

91影视

Find all first and second partial derivatives of \(z\) with respect to \(x\) and \(y\) if \(x^{2}+4 y^{2}+\) \(16 z^{2}-64=0\)

Short Answer

Step by step solution

Identify the Implicit Function

Differentiate Implicitly with Respect to x

Differentiate Implicitly with Respect to y

Find the Second Partial Derivative with Respect to x

Second Partial Derivative with Respect to y

Mixed Second Partial Derivative \( \frac{\partial^2 z}{\partial x \partial y} \)

Mixed Second Partial Derivative \( \frac{\partial^2 z}{\partial y \partial x} \)

Key Concepts

Implicit Differentiation

Second Partial Derivative

Mixed Partial Derivatives

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