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Chapter 13: Problem 41

Find the first partial derivatives of the following functions. $$f(x, y, z)=x y+x z+y z$$

Short Answer

Expert verified
Question: Find the first partial derivatives of the function \(f(x, y, z) = xy + xz + yz\). Answer: The first partial derivatives of the function are as follows: \(\frac{\partial f}{\partial x} = y + z\) \(\frac{\partial f}{\partial y} = x + z\) \(\frac{\partial f}{\partial z} = x + y\)

Step by step solution

01

Find the partial derivative with respect to x

To find the partial derivative with respect to \(x\), we treat \(y\) and \(z\) as constant and differentiate the function \(f(x, y, z)\). Therefore, we will have: $$\frac{\partial f}{\partial x} = \frac{\partial (x y + x z + y z)}{\partial x}$$ Now, apply the derivative on the function: $$\frac{\partial f}{\partial x} = y + z$$
02

Find the partial derivative with respect to y

To find the partial derivative with respect to \(y\), we treat \(x\) and \(z\) as constant and differentiate the function \(f(x, y, z)\). Therefore, we will have: $$\frac{\partial f}{\partial y} = \frac{\partial (x y + x z + y z)}{\partial y}$$ Now, apply the derivative on the function: $$\frac{\partial f}{\partial y} = x + z$$
03

Find the partial derivative with respect to z

To find the partial derivative with respect to \(z\), we treat \(x\) and \(y\) as constant and differentiate the function \(f(x, y, z)\). Therefore, we will have: $$\frac{\partial f}{\partial z} = \frac{\partial (x y + x z + y z)}{\partial z}$$ Now, apply the derivative on the function: $$\frac{\partial f}{\partial z} = x + y$$ In conclusion, the first partial derivatives of the function \(f(x, y, z)=xy+xz+yz\) are: $$\frac{\partial f}{\partial x} = y + z$$ $$\frac{\partial f}{\partial y} = x + z$$ $$\frac{\partial f}{\partial z} = x + y$$

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Most popular questions from this chapter

Find the points (if they exist) at which the following planes and curves intersect. $$8 x+y+z=60 ; \quad \mathbf{r}(t)=\left\langle t, t^{2}, 3 t^{2}\right\rangle, \text { for }-\infty

a. Show that the point in the plane \(a x+b y+c z=d\) nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right),\) where \(D^{2}=a^{2}+b^{2}+c^{2} .\) Conclude that the least distance from the plane to the origin is \(|d| / D\). (Hint: The least distance is along a normal to the plane.) b. Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=d\) is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\).

Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$

What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$
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