Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.83
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
Verified step by step guidance1
Identify the function given: \(y = 3^{\log_{2} t}\). Here, the base of the exponent is 3, and the exponent is \(\log_{2} t\).
Recall that when differentiating an exponential function with a variable exponent, it is often helpful to rewrite the function using the natural exponential and logarithm: \(y = e^{\ln(3^{\log_{2} t})}\).
Use the logarithm power rule to simplify the exponent inside the natural exponential: \(\ln(3^{\log_{2} t}) = \log_{2} t \cdot \ln(3)\), so \(y = e^{\log_{2} t \cdot \ln(3)}\).
Differentiate \(y\) with respect to \(t\) using the chain rule: \(\frac{dy}{dt} = y \cdot \frac{d}{dt} (\log_{2} t \cdot \ln(3))\). Since \(\ln(3)\) is constant, focus on differentiating \(\log_{2} t\).
Recall that \(\log_{2} t = \frac{\ln t}{\ln 2}\), so \(\frac{d}{dt} \log_{2} t = \frac{1}{t \ln 2}\). Substitute this back to get \(\frac{dy}{dt} = 3^{\log_{2} t} \cdot \ln(3) \cdot \frac{1}{t \ln 2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions and Change of Base
Logarithmic functions are the inverses of exponential functions. The change of base formula allows rewriting logarithms with any base into a more convenient base, such as natural logarithms, which simplifies differentiation.
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Change of Base Property
Exponential Functions with Variable Exponents
When the exponent of an exponential function is itself a function of the variable, differentiation requires applying the chain rule and often rewriting the expression using natural exponentials and logarithms.
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Chain Rule in Differentiation
The chain rule is used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
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Related Practice
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