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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 49
Capitolo 3, Problema 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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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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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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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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The Product Rule Example 2
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