Domanda del libro di testo
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
y = xe^y
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.9.47
15–48. Derivatives Find the derivative of the following functions.
f(x) = 2^x/2^x+1
Guida verificata passo dopo passo1
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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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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Derivatives
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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