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Chapter 3: Problem 14

Find the derivative of the following functions. See Example 4 of Section 1 for the derivative of \(\sqrt{x}\). \(g(w)=\frac{5}{6} w^{12}\)

Short Answer

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Question: Find the derivative of the function \(g(w)=\frac{5}{6} w^{12}\). Answer: The derivative of the function is \(g'(w)=10w^{11}\).

Step by step solution

01

Identify the function

The given function is \(g(w)=\frac{5}{6} w^{12}\). We are asked to find its derivative with respect to \(w\).
02

Apply the Power Rule

Applying the power rule to the given function, we take the exponent (12) and multiply it by the coefficient, then subtract 1 from the exponent: Derivative of \(w^{12}\): \((12)w^{12-1} = 12w^{11}\).
03

Multiply by the coefficient

Now, we multiply this result by the coefficient \(\frac{5}{6}\): \(\frac{5}{6}(12w^{11}) = \frac{5 \cdot 12}{6} w^{11}\).
04

Simplify the expression

Simplifying the expression, and cancelling out the common factor of 6, we get: \(g'(w)=\frac{5 \cdot 12}{6} w^{11} = \frac{5 \cdot 2}{1} w^{11} = 10w^{11}\). The derivative of the function \(g(w)=\frac{5}{6} w^{12}\) is \(g'(w)=10w^{11}\).

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Key Concepts

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

Understanding the Power Rule in Differentiation
When it comes to finding the derivative of polynomial functions, the power rule is your go-to tool. This rule is a straightforward method used in calculus to simplify the process of differentiation. Let's break it down:

Given a function of the form \( f(x) = ax^n \), where \(a\) is a constant, \(x\) is the variable, and \(n\) is a real number exponent, the power rule states that the derivative \(f'(x)\) is given by \( f'(x) = nax^{n-1} \). This means you:
  • Multiply the power \(n\) by the coefficient \(a\).
  • Decrease the power \(n\) by one.

This rule efficiently transforms complex polynomials into simpler functions, making it easier to analyze their behavior. By applying this rule to the function \( g(w)=\frac{5}{6} w^{12} \), we quickly calculate its derivative.
Differentiation Explained
Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing at any given point. It's the process of computing a derivative, which represents this rate of change.

In practical terms, differentiation helps to:
  • Determine how the values of a function shift as inputs change.
  • Find slopes of tangent lines to curves, which are crucial in understanding the function's behavior.
  • Optimize functions in real-world applications, such as maximizing profits or minimizing costs.

For the function \( g(w)=\frac{5}{6} w^{12} \), differentiation through the power rule lets us find how \(g\) changes with \(w\), providing a clearer picture of its increasing or decreasing nature.
The Role of Calculus in Analyzing Functions
Calculus, the mathematical study of continuous change, plays a vital role in analyzing and understanding the patterns within functions. It offers powerful techniques for modeling and solving problems that involve change and motion.

Two main branches of calculus are:
  • Differential Calculus: Focuses on finding the derivative, which helps understand how a function evolves over a particular interval.
  • Integral Calculus: Involves finding areas under curves to understand the accumulation of quantities.

By leveraging these concepts, mathematicians and scientists can examine complex systems in physics, engineering, economics, and beyond.
In the context of \( g(w)=\frac{5}{6} w^{12} \), knowing how to differentiate using the power rule is one way calculus provides insight into the dynamics of mathematical models.

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Most popular questions from this chapter

The bottom of a large theater screen is \(3 \mathrm{ft}\) above your eye level and the top of the screen is \(10 \mathrm{ft}\) above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of \(3 \mathrm{ft} / \mathrm{s}\) while looking at the screen. What is the rate of change of the viewing angle \(\theta\) when you are \(30 \mathrm{ft}\) from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Use implicit differentiation if needed to find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2},\) where \(m\) and \(a\) are constants

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}=\frac{x^{2}(4-x)}{4+x} \text { (right strophoid) }$$

One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let \((f g)^{(n)}\) denote the \(n\) th derivative of the product \(f g,\) for \(n \geq 1\) a. Prove that \((f g)^{(2)}=f^{\prime \prime} g+2 f^{\prime} g^{\prime}+f g^{\prime \prime}\) b. Prove that, in general,$$(f g)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\\k\end{array}\right) f^{(k)} g^{(n-k)}$$ where \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\frac{n !}{k !(n-k) !}\) are the binomial coefficients. c. Compare the result of (b) to the expansion of \((a+b)^{n}\).

An angler hooks a trout and begins turning her circular reel at \(1.5 \mathrm{rev} / \mathrm{s}\). If the radius of the reel (and the fishing line on it) is 2 in. then how fast is she reeling in her fishing line?
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