Domanda del libro di testo
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(s) = √s/4
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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 3.9.6
Explain why b^x = e^xlnb.
Guida verificata passo dopo passo1
Start by understanding the concept of exponential functions. The function \( b^x \) is an exponential function where \( b \) is the base and \( x \) is the exponent.
Recall the natural exponential function \( e^x \), which is a fundamental function in calculus due to its unique properties, such as its derivative being itself.
Use the property of logarithms: \( b^x = e^{x \ln b} \). This transformation is based on the identity \( a^b = e^{b \ln a} \), which allows us to express any exponential function in terms of the natural exponential function.
Understand that \( \ln b \) is the natural logarithm of \( b \). The expression \( x \ln b \) is the exponent in the transformed function \( e^{x \ln b} \). This transformation is useful because it allows us to leverage the properties of \( e^x \) in calculus.
Recognize that this transformation is particularly useful in calculus for differentiation and integration, as the derivative of \( e^x \) is \( e^x \), making calculations more straightforward when dealing with exponential functions.
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Exponential Functions
Exponential functions are mathematical expressions in the form of b^x, where b is a positive constant and x is a variable. These functions exhibit rapid growth or decay, depending on the base b. Understanding their properties is crucial for manipulating and transforming exponential expressions.
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6:13
Exponential Functions
Natural Exponential Function
The natural exponential function, denoted as e^x, is a specific exponential function where the base e is approximately equal to 2.71828. It is fundamental in calculus due to its unique property that the derivative of e^x is itself, making it a key function in various applications, including growth models and compound interest.
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04:56Derivative of the Natural Exponential Function (e^x)
Natural Logarithm
The natural logarithm, represented as ln(b), is the logarithm to the base e. It is the inverse operation of the natural exponential function. The relationship between exponentials and logarithms is essential for transforming expressions, as it allows us to express b^x in terms of e, facilitating easier calculations and understanding of growth rates.
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05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
Matching heights A stone is thrown with an initial velocity of 32 ft/s from the edge of a bridge that is 48 ft above the ground. The height of this stone above the ground t seconds after it is thrown is f(t) = −16t²+32t+48 . If a second stone is thrown from the ground, then its height above the ground after t seconds is given by g(t) = −16t²+v0t, where v0 is the initial velocity of the second stone. Determine the value of v0 such that both stones reach the same high point.
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Domanda del libro di testo
Water is drained out of an inverted cone that has the same dimensions as the cone depicted in Exercise 36. If the water level drops at 1 ft/min, at what rate is water (in ft³/min) draining from the tank when the water depth is 6 ft?
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Domanda del libro di testo
5–8. Calculate dy/dx using implicit differentiation.
e^y-e^x = C, where C is constant
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Domanda del libro di testo
Find the derivative of the following functions.
y = In |sin x|
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Domanda del libro di testo
Derivatives Find and simplify the derivative of the following functions.
g(t) = t³+3t²+t / t³
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