Textbook Question
If f′(−2) = 7, find an equation of the line tangent to the graph of f at the point (−2,4).
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 9a
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = (x - 1)(3x + 4)
Verified step by step guidance1
Step 1: Identify the functions to apply the Product Rule. Here, we have two functions: \( u(x) = x - 1 \) and \( v(x) = 3x + 4 \).
Step 2: Recall the Product Rule formula: \( (uv)' = u'v + uv' \). This means we need to find the derivatives of \( u(x) \) and \( v(x) \).
Step 3: Differentiate \( u(x) = x - 1 \). The derivative \( u'(x) \) is 1, since the derivative of \( x \) is 1 and the derivative of a constant is 0.
Step 4: Differentiate \( v(x) = 3x + 4 \). The derivative \( v'(x) \) is 3, since the derivative of \( 3x \) is 3 and the derivative of a constant is 0.
Step 5: Apply the Product Rule: \( f'(x) = u'v + uv' = (1)(3x + 4) + (x - 1)(3) \). Simplify the expression to find the derivative.
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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 you have two functions, u(x) and v(x), the derivative of their product is given by f'(x) = u'v + uv'. This rule is essential when differentiating expressions where two functions are multiplied together, as it allows for the correct application of differentiation principles.
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The Product Rule
Quotient Rule
The Quotient Rule is used to differentiate a function that is the ratio of two other functions. If f(x) = u(x)/v(x), the derivative is given by f'(x) = (u'v - uv')/v^2. This rule is crucial when dealing with fractions of functions, ensuring that the differentiation accounts for both the numerator and denominator appropriately.
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Simplification of Derivatives
Simplification of derivatives involves reducing the expression obtained after differentiation to its simplest form. This may include factoring, combining like terms, or canceling common factors. Simplifying the result is important for clarity and ease of interpretation, especially when further analysis or evaluation of the derivative is required.
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Related Practice
Textbook Question
Let F(x) = f(x) + g(x),G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x), where the graphs of f and g are shown in the figure. Find each of the following.
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H'(2)
1
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Textbook Question
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
g(t) = (t + 1)(t² - t + 1)
Textbook Question
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
h(z) = (z3 + 4z2 + z)(z - 1)
Textbook Question
An equation of the line tangent to the graph of g at x = 3 is y = 5x + 4. Find g(3) and g′(3).
Textbook Question
If h(1) = 2 and h′(1) = 3, find an equation of the line tangent to the graph of h at x = 1.