Domanda del libro di testo
137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.3.59
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
59. y = 2^x
Guida verificata passo dopo passo1
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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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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04:50
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
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