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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 49
Chapter 3, Problem 49

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 
f(w) = w³-w/w

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Step 1: Simplify the expression \( f(w) = \frac{w^3 - w}{w} \) by dividing each term in the numerator by \( w \). This gives \( f(w) = w^2 - 1 \).
Step 2: Recognize that \( f(w) = w^2 - 1 \) is a polynomial function, which is straightforward to differentiate.
Step 3: Apply the power rule for differentiation, which states that \( \frac{d}{dw}[w^n] = nw^{n-1} \).
Step 4: Differentiate each term separately: \( \frac{d}{dw}[w^2] = 2w \) and \( \frac{d}{dw}[-1] = 0 \).
Step 5: Combine the derivatives to find \( f'(w) = 2w \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a given point.
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Derivatives

Product and Quotient Rules

The Product Rule and Quotient Rule are techniques used to find the derivatives of products and quotients of functions, respectively. The Product Rule states that the derivative of two multiplied functions is the first function times the derivative of the second plus the second function times the derivative of the first. The Quotient Rule states that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
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The Quotient Rule

Simplification

Simplification in calculus involves rewriting expressions in a more manageable or understandable form. This can include factoring, expanding, or reducing fractions to make differentiation easier. Simplifying expressions before taking derivatives can often lead to clearer results and can help avoid errors in calculation.
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Related Practice
Textbook Question

Calculate the derivative of the following functions.

g(x) = x / e3x

Textbook Question

Calculate the derivative of the following functions.

y = (1 + 2 tan u)4.5

Textbook Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

f(x) = (√x+1)(√x-1)

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Textbook Question

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

Textbook Question

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.

a. Determine the marginal cost and average cost functions. Graph and interpret these functions.

Textbook Question

The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.

b. Find the velocity function for the marble.