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Chapter 12: Problem 7

Find all first-order partial derivatives. $$f(x, y, z)=3 x \sin y+4 x^{3} y^{2} z$$

Short Answer

Expert verified
The first-order partial derivatives of the given function are \(\frac{\partial f}{\partial x} = 3 \sin y + 12 x^{2} y^{2} z\), \(\frac{\partial f}{\partial y} = 3 x \cos y + 8 x^{3} y z\), and \(\frac{\partial f}{\partial z} = 4 x^{3} y^{2}\).

Step by step solution

01

Partial Derivative with Respect to x

Differentiate \(f(x, y, z)=3 x \sin y+4 x^{3} y^{2} z\) with respect to \(x\), treating \(y\) and \(z\) as constants: \(\frac{\partial f}{\partial x} = 3 \sin y + 12 x^{2} y^{2} z\)
02

Partial Derivative with Respect to y

Next, differentiate \(f(x, y, z)\) with respect to \(y\), treating \(x\) and \(z\) as constants: \(\frac{\partial f}{\partial y} = 3 x \cos y + 8 x^{3} y z\)
03

Partial Derivative with Respect to z

Finally, differentiate \(f(x, y, z)\) with respect to \(z\), treating \(x\) and \(y\) as constants: \(\frac{\partial f}{\partial z} = 4 x^{3} y^{2}\)

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Key Concepts

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

Understanding Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate at which the function changes with respect to one of its variables. In the context of functions like \(f(x, y, z) = 3x \sin y + 4x^3 y^2 z\), differentiation helps us understand how changes in one variable affect the function while keeping others constant.

When we talk about differentiation, it's crucial to know that:
  • The derivative of a constant remains zero.
  • The power rule, which applies to polynomials, helps by reducing the power by one and multiplying by the original power.
  • Trigonometric functions have their distinct rules, such as \(\frac{d}{dx}(\sin y) = \cos y\).
For multivariable functions, correct identification of each variable is essential to apply the right differentiation techniques.
Exploring Multivariable Calculus
Multivariable calculus extends single-variable calculus concepts to functions with more than one variable. Here, the function \(f(x, y, z)\) depends on three variables \(x\), \(y\), and \(z\). Rather than considering the function's change along a single axis, multivariable calculus assesses changes in multiple dimensions.

Key points to remember:
  • Functions can have derivatives with respect to each variable independently.
  • This branch of calculus helps model real-world problems where outcomes depend on several factors.
  • Visualization often involves using surfaces instead of curves.
Grasping multivariable calculus is crucial for fields like physics and engineering, where systems are frequently modeled using multiple influencing factors.
Introducing First-Order Partial Derivatives
First-order partial derivatives represent the rate of change of a multivariable function concerning one variable, with the others held constant. For instance, finding the partial derivative \(\frac{\partial f}{\partial x}\) of \(f(x, y, z)\) involves treating \(y\) and \(z\) as constants while differentiating with respect to \(x\).

Here's how partial derivatives provide insight:
  • They pinpoint how each variable directly affects the function independently of others.
  • The process involves standard differentiation techniques, as seen in single-variable calculus but applied to one variable at a time.
  • These derivatives are crucial for optimization problems where we need to understand each variable's impact independently.
Understanding partial derivatives forms the foundation for more advanced topics, like gradient and Hessian, used in deeper economic or scientific analyses.

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Most popular questions from this chapter

A well-known college uses the following formula to predict the grade average of prospective students: \(\mathrm{PGA}=0.708(\mathrm{HS})+0.0018(\mathrm{SATV})+0.001(\mathrm{SATM})-1.13\) Here, PGA is the predicted grade average, HS is the student's high school grade average (in core academic courses, on a four point scale), SATV is the student's SAT verbal score and SATM is the student's SAT math score. Use your scores to compute your own predicted grade average. Determine whether it is possible to have a predicted average of \(4.0,\) or a negative predicted grade average. In this formula, the predicted grade average is a function of three variables. State which variable you think is the most important and explain why you think so.

Calculate one step of the steepest ascent algorithm for \(f(x, y)=2 x y-2 x^{2}+y^{3},\) starting at \((0,0) .\) Explain in graphical terms what goes wrong.

Prove that the situation of exercise 66 (two local minima without a local maximum) can never occur for differentiable functions of one variable.

At a certain point on a mountain, a surveyor sights due east and measures a \(10^{\circ}\) drop-off, then sights due north and measures a \(6^{\circ}\) rise. Find the direction of steepest ascent and compute the degree rise in that direction.

For a rectangle of length \(L\) and perimeter \(P\), show that the area is given by \(A=\frac{1}{2} L P-L^{2} .\) Compute \(\frac{\partial A}{\partial L} .\) A simpler formula for area is \(A=L W,\) where \(W\) is the width of the rectangle. Compute \(\frac{\partial A}{\partial L}\) and show that your answer is not equivalent to the previous derivative. Explain the difference by noting that in one case the width is held constant while \(L\) changes, whereas in the other case the perimeter is held constant while \(L\) changes.
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