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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 3.4.46
Chapter 3, Problem 3.4.46

Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)

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Step 1: Identify the function as a quotient of two functions, where the numerator is \((x-1)(2x^2-1)\) and the denominator is \(x^3-1\). We will use the quotient rule for derivatives, which is given by \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\), where \(u\) is the numerator and \(v\) is the denominator.
Step 2: Differentiate the numerator \(u = (x-1)(2x^2-1)\) using the product rule. The product rule states that \(\frac{d}{dx}(fg) = f'g + fg'\). Let \(f(x) = x-1\) and \(g(x) = 2x^2-1\). Find \(f'(x)\) and \(g'(x)\), then apply the product rule.
Step 3: Differentiate the denominator \(v = x^3-1\). The derivative of \(v\) is \(v' = 3x^2\).
Step 4: Substitute \(u\), \(u'\), \(v\), and \(v'\) into the quotient rule formula. Simplify the expression obtained from the quotient rule.
Step 5: Simplify the resulting expression further, if possible, by combining like terms and reducing any common factors in the numerator and the denominator.

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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 at which a function changes at a given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, derivatives are fundamental for understanding the behavior of functions, including their slopes and rates of change.
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Derivatives

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as seen in the function h(x) provided.
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Quotient Rule

The Quotient Rule is used to differentiate functions that are expressed as the ratio of two other functions. If h(x) = u(x)/v(x), the derivative is given by (u'v - uv')/v². This rule is crucial for the function h(x) in the question, as it involves a division of two polynomial expressions.
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Related Practice
Textbook Question

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

Textbook Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

Textbook Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

Textbook Question

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

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Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

Textbook Question

Evaluate the derivative of the following functions.

f(x) = sin-1 2x