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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.96c
Chapter 3, Problem 3.96c

Computing the derivative of f(x) = e^-x
c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Verified step by step guidance
1
Step 1: Recall the derivative of the exponential function. The derivative of \( e^x \) with respect to \( x \) is \( e^x \).
Step 2: Apply the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).
Step 3: Identify the inner function \( g(x) = -x \) and the outer function \( f(u) = e^u \) where \( u = g(x) \).
Step 4: Differentiate the inner function \( g(x) = -x \). The derivative \( g'(x) \) is \( -1 \).
Step 5: Combine the results using the chain rule. The derivative of \( f(x) = e^{-x} \) is \( e^{-x} \cdot (-1) \).

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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. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.
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Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. The function f(x) = e^-x is a specific case where the base 'e' is raised to the power of a negative variable, resulting in a function that decreases rapidly as x increases. Understanding the properties of exponential functions is essential for computing their derivatives.
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Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions like e^-x, where the exponent itself is a function of x.
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Related Practice
Textbook Question

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c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

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Textbook Question

Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

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Textbook Question

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c. Find the average velocity of the car over the interval [1.75, 2.25]. Estimate the velocity of the car at 11:00 A.M. and determine the direction in which the patrol car is moving.

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Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=e^−x tan^−1 x on [0,∞)

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Textbook Question

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c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.

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Textbook Question

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73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

c. f(x) = √|x-4|

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