Textbook Question
Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).
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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 3.3.7
Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).
Verified step by step guidance1
Step 1: Understand the problem. We are given the derivatives of two functions, f and g, at x = 3. Specifically, f'(3) = 6 and g'(3) = -2. We need to find the derivative of the sum of these functions, (f+g)'(3).
Step 2: Recall the rule for the derivative of a sum. The derivative of the sum of two functions is the sum of their derivatives. Mathematically, this is expressed as (f+g)'(x) = f'(x) + g'(x).
Step 3: Apply the rule to the given point. Substitute x = 3 into the formula: (f+g)'(3) = f'(3) + g'(3).
Step 4: Substitute the given values. We know f'(3) = 6 and g'(3) = -2, so substitute these values into the equation: (f+g)'(3) = 6 + (-2).
Step 5: Simplify the expression. Combine the values to find the derivative of the sum at x = 3.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of a Function
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is denoted as f'(x) and represents the slope of the tangent line to the function's graph at that point. Understanding derivatives is crucial for analyzing how functions behave locally.
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Derivatives of Other Trig Functions
Sum Rule of Derivatives
The Sum Rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if f and g are functions, then (f + g)'(x) = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with combined functions.
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Evaluating Derivatives at a Point
To evaluate the derivative of a function at a specific point, you substitute the point's value into the derivative function. In this case, knowing f'(3) and g'(3) allows us to find (f + g)'(3) by directly applying the Sum Rule and substituting the given values.
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Related Practice
Textbook Question
49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
f (x) = (sin x)^In x; a = π/2
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Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).
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