Textbook Question
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
(g^-1)'(7)
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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 87
Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>
d/dx (f(x)g(x)) | x=4
Verified step by step guidance1
Step 1: Recall the product rule for derivatives, which states that if you have two functions f(x) and g(x), the derivative of their product is given by \( \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \).
Step 2: Identify the values of f(x), g(x), f'(x), and g'(x) at x = 4 from the graph. You will need to find the y-values of f(x) and g(x) at x = 4, as well as the slopes of the tangent lines to f(x) and g(x) at x = 4.
Step 3: Substitute the values of f(x), g(x), f'(x), and g'(x) at x = 4 into the product rule formula. This will give you the expression for the derivative of the product at x = 4.
Step 4: Simplify the expression obtained in Step 3 to find the derivative of the product at x = 4.
Step 5: Verify your result by checking the calculations and ensuring that the values from the graph are correctly interpreted.
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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 fundamental principle in calculus used to differentiate the product of two functions. It states that if you have two functions, f(x) and g(x), the derivative of their product is given by f'(x)g(x) + f(x)g'(x). This rule is essential for finding the derivative of the expression d/dx (f(x)g(x)) at a specific point, such as x=4.
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The Product Rule
Evaluating Derivatives at a Point
Evaluating derivatives at a specific point involves substituting the value of x into the derivative function obtained from differentiation. In this case, after applying the Product Rule, you will need to calculate the values of f(4), g(4), f'(4), and g'(4) to find the derivative at x=4. This step is crucial for obtaining a numerical answer.
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Graphical Interpretation of Derivatives
The graphical interpretation of derivatives involves understanding how the slope of the tangent line to a curve at a given point represents the derivative at that point. By analyzing the graph of f(x) and g(x), one can visually assess the behavior of the functions and their derivatives, which aids in comprehending the results obtained through algebraic differentiation.
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Related Practice
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
(4x+1)ln x = xln(4x+1)
Textbook Question
Second derivatives Find d²y/dx²for the following functions.
y = x cos x²
Textbook Question
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
d. d/dx (f(x)³) |x=5
Textbook Question
Second derivatives Find d²y/dx²for the following functions.
y = √x²+2
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
d/dx((√2)x) = x(√2)x - 1