Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.9.6

Chapter 3, Problem 3.9.6

Explain why b^x = e^xlnb.

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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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Key Concepts

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

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.

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.

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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Derivative of the Natural Logarithmic Function

Related Practice

Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

5–8. Calculate dy/dx using implicit differentiation.

e^y-e^x = C, where C is constant

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Textbook Question

Find the derivative of the following functions.

y = In |sin x|

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