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Chapter 3: Problem 47

Find the indicated higher-order partial derivatives. Let \(F(x, y, z)=x^{3} y z^{2}-2 x^{2} y z+3 x z-2 y^{3} z .\) Find \(F_{x y z}\)

Short Answer

Expert verified
The higher-order partial derivative \(F_{xyz} = 6x^2 z - 4x\).

Step by step solution

01

Find the First Partial Derivative with respect to x

To find the partial derivative \(F_x\), treat \(y\) and \(z\) as constants. Differentiate \(F(x, y, z)\) with respect to \(x\):\[ F_x = \frac{\partial}{\partial x}(x^{3} y z^{2} - 2x^{2} y z + 3xz - 2y^{3} z) \]- The derivative of \(x^3 y z^2\) with respect to \(x\) is \(3x^2 y z^2\).- The derivative of \(-2x^2 y z\) with respect to \(x\) is \(-4x y z\).- The derivative of \(3xz\) with respect to \(x\) is \(3z\).- The derivative of \(-2y^3 z\) with respect to \(x\) is \(0\) (since \(-2y^3 z\) does not contain \(x\)).So,\[ F_x = 3x^2 y z^2 - 4x y z + 3z \]
02

Find the Second Partial Derivative with respect to y

Now, differentiate \(F_x\) with respect to \(y\):\[ F_{xy} = \frac{\partial}{\partial y}(3x^2 y z^2 - 4x y z + 3z) \]- The derivative of \(3x^2 y z^2\) with respect to \(y\) is \(3x^2 z^2\).- The derivative of \(-4x y z\) with respect to \(y\) is \(-4x z\).- The derivative of \(3z\) with respect to \(y\) is \(0\).Thus,\[ F_{xy} = 3x^2 z^2 - 4x z \]
03

Find the Third Partial Derivative with respect to z

Finally, differentiate \(F_{xy}\) with respect to \(z\):\[ F_{xyz} = \frac{\partial}{\partial z}(3x^2 z^2 - 4x z) \]- The derivative of \(3x^2 z^2\) with respect to \(z\) is \(6x^2 z\).- The derivative of \(-4x z\) with respect to \(z\) is \(-4x\).Therefore,\[ F_{xyz} = 6x^2 z - 4x \]

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Key Concepts

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

Understanding Partial Derivatives with Respect to x
A partial derivative with respect to a specific variable, such as \( x \), involves differentiating a function while treating all other variables as constants. In this process, we're focusing on how the function changes along the \( x \) direction, considering \( y \) and \( z \) as fixed.

Let's apply this to the function \( F(x, y, z) = x^3 y z^2 - 2x^2 y z + 3xz - 2y^3 z \):
  • For the term \( x^3 y z^2 \), we differentiate with respect to \( x \), considering \( y z^2 \) as a constant multiplier. The derivative is \( 3x^2 y z^2 \).
  • The term \( -2x^2 y z \) becomes \( -4x y z \) after differentiation, where \( 2y z \) is treated as a constant.
  • The \( 3xz \) term simplifies to \( 3z \) after differentiating, as \( z \) is constant: we only focus on \( x \).
  • Finally, \( -2y^3 z \) does not contain \( x \), resulting in a derivative of \( 0 \).
Combining these, the partial derivative with respect to \( x \) is \( F_x = 3x^2 y z^2 - 4x y z + 3z \).

Understanding how the function changes with \( x \) helps see its impact along the \( x \)-axis. This is the first step in relating partial derivatives in multivariable functions.
Exploring Partial Derivatives with Respect to y
To find the partial derivative with respect to \( y \), we treat all other variables as constants, just as we did with \( x \). We aim to understand how changes in \( y \) alone impact the function.

Starting from the partial derivative \( F_x = 3x^2 y z^2 - 4x y z + 3z \), we differentiate with respect to \( y \):
  • The term \( 3x^2 y z^2 \) becomes \( 3x^2 z^2 \) after differentiation. Here, \( x^2 z^2 \) is treated as a constant.
  • For \( -4x y z \), the partial derivative with respect to \( y \) is \( -4x z \), considering \( 4x z \) constant with respect to \( y \).
  • The constant \( 3z \) vanishes in differentiation since it's independent of \( y \).
Thus, the partial derivative concerning \( y \) results in \( F_{xy} = 3x^2 z^2 - 4x z \).

This step highlights how variations in \( y \) influence the function, holding \( x \) and \( z \) steady. This insight can reveal patterns that arise solely due to shifts along the \( y \)-axis.
Discovering Partial Derivatives with Respect to z
Differentiating with respect to \( z \), we focus on how \( z \) alone affects the function while considering \( x \) and \( y \) as constants. This will complete our understanding of higher-order partial derivatives.

Using \( F_{xy} = 3x^2 z^2 - 4x z \), we differentiate with respect to \( z \):
  • For \( 3x^2 z^2 \), the derivative is \( 6x^2 z \). Here, \( x^2 \) is constant, and the power rule applies to the \( z^2 \) term.
  • The term \( -4x z \), once differentiated with respect to \( z \), yields \( -4x \).
Gathering these results, the partial derivative with respect to \( z \) is \( F_{xyz} = 6x^2 z - 4x \).

This derivative helps us understand the impact of changes in \( z \) on the function. By piecing together these partial derivatives, we compile a comprehensive view of how changes in each individual variable alter the entire function.

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Most popular questions from this chapter

A large container in the shape of a rectangular solid must have a volume of \(480 \mathrm{~m}^{3}\). The bottom of the container costs \(\$ 5 / \mathrm{m}^{2}\) to construct whereas the top and sides cost \(\$ 3 / \mathrm{m}^{2}\) to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has the minimum cost.

Find the equation for the tangent plane to the surface at the indicated point. $$ z=e^{7 x^{2}+4 y^{2}}, P(0,0,1) $$

For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. $$ f(x, y)=x y ; 4 x^{2}+8 y^{2}=16 $$

For the following exercises, find the equation of the tangent plane to the specified surface at the given point. $$ z=x^{3}-2 y^{2}+y-1 \text { at point }(1,1,-1) $$

Find the equation for the tangent plane to the surface at the indicated point. $$ z=\ln \left(10 x^{2}+2 y^{2}+1\right), P(0,0,0) $$
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