Domanda del libro di testo
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = (x - 1)(3x + 4)
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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 8a
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
g(t) = (t + 1)(t² - t + 1)
Guida verificata passo dopo passo1
Identify the function as a product of two functions: \( u(t) = t + 1 \) and \( v(t) = t^2 - t + 1 \).
Recall the Product Rule for derivatives, which states: \( (uv)' = u'v + uv' \).
Find the derivative of \( u(t) = t + 1 \), which is \( u'(t) = 1 \).
Find the derivative of \( v(t) = t^2 - t + 1 \), which is \( v'(t) = 2t - 1 \).
Apply the Product Rule: \( g'(t) = u'(t)v(t) + u(t)v'(t) = 1(t^2 - t + 1) + (t + 1)(2t - 1) \). Simplify the expression to get the final derivative.
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Product Rule
The Product Rule is a formula used to find the derivative of the product of two functions. If you have two functions, u(t) and v(t), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating expressions where two functions are multiplied together, as it allows for a systematic approach to finding the derivative.
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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 f(t) = u(t)/v(t), the derivative is given by (u'v - uv')/v². This rule is particularly useful when dealing with fractions of functions, ensuring that the differentiation accounts for both the numerator and the denominator.
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06:43The Quotient Rule
Simplification of Derivatives
After applying the Product or Quotient Rule, the resulting derivative often needs simplification to make it more manageable or interpretable. This involves combining like terms, factoring, or reducing fractions. Simplification is crucial for clearer analysis and understanding of the behavior of the function, especially when evaluating limits or finding critical points.
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Percorso guidato
05:44Derivatives
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