Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 14a

Chapter 3, Problem 14a

7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
y = x² - 2ax +a² / x-a, where a is a constant

Verified step by step guidance

Step 1: Identify the function as a quotient of two functions, where the numerator is \( u(x) = x^2 - 2ax + a^2 \) and the denominator is \( v(x) = x - a \).

Step 2: Recall the Quotient Rule for derivatives, which states that if \( y = \frac{u(x)}{v(x)} \), then \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).

Step 3: Compute the derivative of the numerator, \( u'(x) = \frac{d}{dx}(x^2 - 2ax + a^2) = 2x - 2a \).

Step 4: Compute the derivative of the denominator, \( v'(x) = \frac{d}{dx}(x - a) = 1 \).

Step 5: Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the Quotient Rule formula and simplify the expression.

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Key Concepts

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

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the derivative of their product is given by (u*v)' = u'v + uv'. This rule is essential when dealing with expressions where two functions are multiplied together, allowing for the correct application of differentiation.

Quotient Rule

The Quotient Rule is used to differentiate a function that is the quotient of two other functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient is given by (u/v)' = (u'v - uv') / v². This rule is particularly important when the function is expressed as a fraction, ensuring that the differentiation accounts for both the numerator and denominator.

Simplification of Derivatives

Simplification of derivatives involves reducing the expression obtained after differentiation to its simplest form. This may include factoring, canceling common terms, or combining like terms. Simplifying the derivative is crucial for clarity and ease of interpretation, especially when analyzing the behavior of the function or finding critical points.

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Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

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Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

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y = (3x+7)¹⁰

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Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

y = x² - a² / x-a, where a is a constant