Domanda del libro di testo
Evaluate the integrals in Exercises 31–78.
55. ∫(from -2 to -1)e^(-(x+1)) 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.PE.1
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
1. y = 10e^(-x/5)
Guida verificata passo dopo passo1
Identify the function given: \(y = 10e^{-\frac{x}{5}}\).
Recall the derivative rule for exponential functions: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot u'(x)\).
Set the inner function \(u(x) = -\frac{x}{5}\) and find its derivative: \(u'(x) = -\frac{1}{5}\).
Apply the chain rule: \(\frac{dy}{dx} = 10 \cdot e^{-\frac{x}{5}} \cdot \left(-\frac{1}{5}\right)\).
Simplify the expression by multiplying constants to express the derivative clearly.
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Derivative of Exponential Functions
The derivative of an exponential function with base e, such as e^u, is found by multiplying e^u by the derivative of the exponent u. This uses the chain rule and reflects how exponential growth or decay rates change with respect to the variable.
Video consigliato:
04:50
Derivatives of General Exponential Functions
Chain Rule
The chain rule is a method for differentiating composite functions. When a function is composed of an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Video consigliato:
05:02Intro to the Chain Rule
Constant Multiple Rule
The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This simplifies differentiation when constants are involved.
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04:02The Power Rule
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