Domanda del libro di testo
A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.5.32
23–51. Calculating derivatives Find the derivative of the following functions.
y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants
Guida verificata passo dopo passo1
Step 1: Identify the function for which you need to find the derivative. The function given is \( y = \frac{a \sin x + b \cos x}{a \sin x - b \cos x} \).
Step 2: Recognize that this is a quotient of two functions, so you will need to use the Quotient Rule for differentiation. The Quotient Rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then the derivative \( y' \) is given by \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
Step 3: Identify \( u(x) = a \sin x + b \cos x \) and \( v(x) = a \sin x - b \cos x \). Compute the derivatives \( u'(x) \) and \( v'(x) \). Use the derivatives of sine and cosine: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).
Step 4: Calculate \( u'(x) = a \cos x - b \sin x \) and \( v'(x) = a \cos x + b \sin x \).
Step 5: Substitute \( u(x) \), \( v(x) \), \( u'(x) \), and \( v'(x) \) into the Quotient Rule formula to find the derivative of \( y \). Simplify the expression if possible.
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Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Percorso guidato
05:44
Derivatives
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In calculus, these functions are essential for modeling oscillatory behavior and are frequently encountered in problems involving derivatives. Understanding their properties, such as their derivatives, is crucial for solving calculus problems involving these functions.
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6:04Introduction to Trigonometric Functions
Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. It states that if you have a function y = u/v, where u and v are both differentiable functions, the derivative is given by (v * u' - u * v') / v^2. This rule is particularly useful when differentiating functions like the one in the question, where the function is expressed as a fraction.
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06:43The Quotient Rule
Pratica correlata
Domanda del libro di testo
15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π
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Domanda del libro di testo
Find the derivative of the following functions.
y = x sin x
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Domanda del libro di testo
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=4e^10x; (4,0)
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Domanda del libro di testo
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
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Domanda del libro di testo
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = e^3x+1
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