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Chapter 12: Problem 52

Find the first partial derivatives of the following functions. $$g(w, x, y, z)=\cos (w+x) \sin (y-z)$$

Short Answer

Expert verified
Answer: The first partial derivatives of the function are: $$\frac{\partial g}{\partial w} = - \sin (w+x) \sin (y-z)$$ $$\frac{\partial g}{\partial x} = - \sin (w+x) \sin (y-z)$$ $$\frac{\partial g}{\partial y} = \cos (w+x) \cos (y-z)$$ $$\frac{\partial g}{\partial z} = - \cos (w+x) \cos (y-z)$$

Step by step solution

01

Differentiate g with respect to w

To find the partial derivative of g with respect to w, we keep the other variables (x, y, and z) constant and differentiate with respect to w. It is important to recall the derivative of \(\cos(z)\) is \(-\sin(z)\) and the derivative of \(\sin(z)\) is \(\cos(z)\). Applying the chain rule, the partial derivative with respect to w is: \(\frac{\partial g}{\partial w} = - \sin (w+x) \sin (y-z)\)
02

Differentiate g with respect to x

Keeping w, y, and z constant and differentiating with respect to x, we obtain the partial derivative with respect to x: \(\frac{\partial g}{\partial x} = - \sin (w+x) \sin (y-z) \)
03

Differentiate g with respect to y

Keeping w, x, and z constant and differentiating with respect to y, we obtain the partial derivative with respect to y: \(\frac{\partial g}{\partial y} = \cos (w+x) \cos (y-z) \)
04

Differentiate g with respect to z

Keeping w, x, and y constant and differentiating with respect to z, we obtain the partial derivative with respect to z: \(\frac{\partial g}{\partial z} = - \cos (w+x) \cos (y-z) \) The first partial derivatives of the function \(g(w, x, y, z)\) are: $$\frac{\partial g}{\partial w} = - \sin (w+x) \sin (y-z)$$ $$\frac{\partial g}{\partial x} = - \sin (w+x) \sin (y-z)$$ $$\frac{\partial g}{\partial y} = \cos (w+x) \cos (y-z)$$ $$\frac{\partial g}{\partial z} = - \cos (w+x) \cos (y-z)$$

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Most popular questions from this chapter

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{-x} \sin y$$

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 \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\) (Hint: Find the point \(P\) on the plane closest to \(P_{0}\).)

Let \(x, y,\) and \(z\) be non-negative numbers with \(x+y+z=200\) a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\) b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\) d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).

Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of \(R,\) where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). a. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for \(R\) and differentiating. b. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. c. Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects \(R\). d. Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects \(R\).

Find the points at which the plane \(a x+b y+c z=d\) intersects the \(x-y-\), and \(z\) -axes.
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