Domanda del libro di testo
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x) = (x−1) sin^−1 x on [−1,1]
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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 3.4.12a
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
g(s) = 4s³ - 8s² +4s / 4s
Guida verificata passo dopo passo1
Step 1: Identify the function g(s) = \(\frac{4s^3 - 8s^2 + 4s}{4s}\). This is a rational function, so we will use the Quotient Rule to find its derivative.
Step 2: Recall the Quotient Rule, which states that if you have a function h(s) = \(\frac{u(s)}{v(s)}\), then the derivative h'(s) is given by \(\frac{u'(s)v(s) - u(s)v'(s)}{(v(s))^2}\).
Step 3: Assign u(s) = 4s^3 - 8s^2 + 4s and v(s) = 4s. Calculate the derivatives: u'(s) = \(\frac{d}{ds}\)(4s^3 - 8s^2 + 4s) and v'(s) = \(\frac{d}{ds}\)(4s).
Step 4: Compute u'(s) = 12s^2 - 16s + 4 and v'(s) = 4. Substitute these into the Quotient Rule formula: g'(s) = \(\frac{(12s^2 - 16s + 4)(4s) - (4s^3 - 8s^2 + 4s)(4)}{(4s)^2}\).
Step 5: Simplify the expression for g'(s) by expanding the terms in the numerator and then combining like terms. Finally, simplify the entire expression by dividing each term by (4s)^2.
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Product Rule
The Product Rule is a formula used to find the derivative of the product of two functions. If u(s) and v(s) are two differentiable functions, the derivative of their product is given by u'v + uv'. This rule is essential when dealing with functions that are multiplied together, allowing for the correct application of differentiation.
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05:18
The Product Rule
Quotient Rule
The Quotient Rule is used to differentiate a function that is the quotient of two other functions. If u(s) and v(s) are differentiable functions, the derivative of their quotient is given by (u'v - uv') / v². This rule is particularly important when the function is expressed as a fraction, ensuring accurate differentiation while considering the relationship between the numerator and denominator.
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06:43The Quotient Rule
Simplification of Derivatives
After applying the Product or Quotient Rule, it is often necessary to simplify the resulting expression. This involves combining like terms, factoring, or reducing fractions to make the derivative easier to interpret and use. Simplification is a crucial step in calculus, as it helps clarify the behavior of the function and its rate of change.
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Percorso guidato
05:44Derivatives
Pratica correlata
Domanda del libro di testo
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(s) = 4s³+3s; a= -3, -1
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Domanda del libro di testo
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)
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Domanda del libro di testo
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 1/x; P (1,1)
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Domanda del libro di testo
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
a. f(x) = (x-2)^1/3
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Domanda del libro di testo
Deriving trigonometric identities
a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.
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