Textbook Question
Calculate the derivative of the following functions.
g(x) = x / e3x
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 53
Calculate the derivative of the following functions.
y = sin(sin(ex))
Verified step by step guidance1
Step 1: Identify the outermost function and the inner functions. Here, the outermost function is \( \sin(u) \), where \( u = \sin(v) \) and \( v = e^x \).
Step 2: Apply the chain rule. The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
Step 3: Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \).
Step 4: Differentiate the middle function \( \sin(v) \) with respect to \( v \). The derivative is \( \cos(v) \).
Step 5: Differentiate the innermost function \( e^x \) with respect to \( x \). The derivative is \( e^x \).
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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 a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when dealing with nested functions, such as in the given problem where sine functions are composed with an exponential function.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. In calculus, these functions have specific derivatives: the derivative of sin(x) is cos(x). Understanding the behavior and derivatives of these functions is crucial when calculating derivatives of more complex expressions involving trigonometric functions.
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Related Practice
Textbook Question
Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
c. Sketch a graph of f'.
Textbook Question
Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
b. Find the values of x in (0, 3) at which f is not differentiable.
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
{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
Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (0, 3) at which f is not continuous.