Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
25. y = ln(ln(x))
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.59
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
59. y = 2^x
Verified step by step guidance1
Recognize that the function is an exponential function of the form \(y = a^x\) where \(a = 2\) is a constant base and \(x\) is the variable.
Recall the general formula for the derivative of an exponential function with a constant base: \(\frac{d}{dx} a^x = a^x \ln(a)\).
Apply this formula to the given function by substituting \(a = 2\), so the derivative becomes \(\frac{dy}{dx} = 2^x \ln(2)\).
Understand that \(\ln(2)\) is the natural logarithm of 2, which is a constant multiplier in the derivative.
Write the final expression for the derivative as \(\frac{dy}{dx} = 2^x \ln(2)\), which completes the differentiation process.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Exponential Functions
The derivative of an exponential function with a constant base, such as y = a^x, involves the natural logarithm of the base. Specifically, the derivative is dy/dx = a^x * ln(a), where ln(a) is the natural log of the base a. This rule helps differentiate functions like y = 2^x.
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Derivatives of General Exponential Functions
Natural Logarithm (ln)
The natural logarithm, denoted ln(x), is the inverse of the exponential function with base e. It is essential in differentiation of exponential functions with bases other than e, as it appears in the derivative formula to account for the base's growth rate.
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05:18Derivative of the Natural Logarithmic Function
Basic Differentiation Rules
Understanding basic differentiation rules, such as the power rule and constant multiple rule, is crucial. These rules allow you to differentiate functions systematically and combine with the exponential derivative formula to find the derivative of more complex expressions.
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04:00Solutions to Basic Differential Equations
Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
7. lim (x → 2) (x - 2) / (x² - 4)
Textbook Question
Solve the differential equation in Exercises 9–22.
11. (dy/dx) = e^(x-y)
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Textbook Question
133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed.
Textbook Question
Solve the differential equation in Exercises 9–22.
15. √x (dy/dx) = e^(y+√x), x > 0
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
69. y = 2^(sin 3t)