Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
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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 7.3.7
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(-5x)
Guida verificata passo dopo passo1
Identify the function given: \(y = e^{-5x}\), where \(y\) is expressed in terms of \(x\).
Recall the chain rule for differentiation: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}\).
Set the inner function \(u(x) = -5x\) and find its derivative with respect to \(x\): \(\frac{du}{dx} = -5\).
Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: \(\frac{dy}{dx} = e^{-5x} \cdot (-5)\).
Write the final expression for the derivative as \(\frac{dy}{dx} = -5 e^{-5x}\) (do not calculate the numerical value).
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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 rule is essential for differentiating functions like y = e^(-5x), where the exponent is a function of x.
Video consigliato:
04:50
Derivatives of General Exponential Functions
Chain Rule
The chain rule is used to differentiate 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. For y = e^(-5x), the inner function is -5x.
Video consigliato:
05:02Intro to the Chain Rule
Notation and Variable Dependence
Understanding which variable to differentiate with respect to (x, t, or θ) is crucial. In this problem, y is expressed in terms of x, so differentiation is with respect to x. Recognizing the independent variable ensures correct application of differentiation rules.
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Percorso guidato
04:22Sigma Notation
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