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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 3.9.47
Chapter 3, Problem 3.9.47

15–48. Derivatives Find the derivative of the following functions.
f(x) = 2^x/2^x+1

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Step 1: Identify the function f(x) = \( \frac{2^x}{2^x + 1} \). This is a quotient of two functions, so we will use the quotient rule to find the derivative.
Step 2: Recall the quotient rule for derivatives: If you have a function \( g(x) = \frac{u(x)}{v(x)} \), then the derivative \( g'(x) \) is given by \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
Step 3: Assign \( u(x) = 2^x \) and \( v(x) = 2^x + 1 \). Now, find the derivatives \( u'(x) \) and \( v'(x) \).
Step 4: The derivative of \( u(x) = 2^x \) is \( u'(x) = 2^x \ln(2) \) because the derivative of \( a^x \) is \( a^x \ln(a) \). The derivative of \( v(x) = 2^x + 1 \) is \( v'(x) = 2^x \ln(2) \) since the derivative of a constant is zero.
Step 5: Substitute \( u(x), u'(x), v(x), \) and \( v'(x) \) into the quotient rule formula: \( f'(x) = \frac{2^x \ln(2) (2^x + 1) - 2^x (2^x \ln(2))}{(2^x + 1)^2} \). Simplify the expression to find the derivative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function's graph at any given point. Understanding how to compute derivatives is essential for analyzing the behavior of functions, including their increasing or decreasing nature.
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Quotient Rule

The Quotient Rule is a formula used to find the derivative of a function that is the ratio of two other functions. If you have a function f(x) = g(x)/h(x), the derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is crucial for differentiating functions like the one in the question, where the function is expressed as a fraction.
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Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. They are characterized by their rapid growth or decay and have unique properties, such as the derivative f'(x) = a^x ln(a). In the context of the given function, understanding how to differentiate exponential terms is vital for correctly applying the rules of calculus.
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Related Practice
Textbook Question

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

y = xe^y

Textbook Question

23–51. Calculating derivatives Find the derivative of the following functions.

y = sin x / 1 + cos x

Textbook Question

If f is continuous at a, must f be differentiable at a?

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) = (4 sin x+2)^cos x; a = π

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Textbook Question

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

Suppose the speed of a car approaching a stop sign is given by v (t) = (t-5)², for 0 ≤ t ≤ 5, where t is measured in seconds and v(t) is measured in meters per second. Find v′(3).

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(x) = 6x⁵ - 5/2 x² + x + 5

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