Textbook Question
Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>
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 50
Calculate the derivative of the following functions.
g(x) = x / e3x
Verified step by step guidance1
Step 1: Identify the function g(x) = \(\frac{x}{e^{3x}\)} as a quotient of two functions, where the numerator is f(x) = x and the denominator is h(x) = e^{3x}.
Step 2: Recall the Quotient Rule for derivatives, which states that if you have a function g(x) = \(\frac{f(x)}{h(x)}\), then its derivative g'(x) is given by \(\frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}\).
Step 3: Calculate the derivative of the numerator, f'(x). Since f(x) = x, its derivative is f'(x) = 1.
Step 4: Calculate the derivative of the denominator, h'(x). Since h(x) = e^{3x}, use the chain rule to find h'(x) = 3e^{3x}.
Step 5: Substitute f(x), f'(x), h(x), and h'(x) into the Quotient Rule formula to find g'(x) = \(\frac{1 \cdot e^{3x}\) - x \(\cdot\) 3e^{3x}}{(e^{3x})^2}.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.
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Derivatives
Quotient Rule
The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. If you have a function g(x) = u(x) / v(x), the derivative g'(x) is given by (u'v - uv') / v², where u' and v' are the derivatives of u and v, respectively. This rule is essential when differentiating functions that are expressed as fractions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828), and a and b are constants. The derivative of an exponential function is unique because it is proportional to the function itself, meaning that d/dx(e^(bx)) = b * e^(bx). Understanding how to differentiate exponential functions is crucial when they appear in more complex expressions.
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Related Practice
Textbook Question
An angler hooks a trout and begins turning her circular reel at 1.5 rev/s. Assume the radius of the reel (and the fishing line on it) is 2 inches.
a. Let R equal the number of revolutions the angler has turned her reel and suppose L is the amount of line that she has reeled in. Find an equation for L as a function of R.
Textbook Question
Calculate the derivative of the following functions.
y = sin(sin(ex))
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Textbook Question
{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.
a. Determine the marginal cost and average cost functions. Graph and interpret these functions.
Textbook Question
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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Textbook Question
The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.
b. Find the velocity function for the marble.