Multiple Choice
For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
4:06 minutesProblem 3.9.47
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
f(x) = 2^x/2^x+1
Verified step by step guidance1
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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