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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 3.3.61
Capitolo 3, Problema 3.3.61

Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

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The Derivative Product Rule states that if you have two functions u(x) and v(x), the derivative of their product is given by: (u*v)' = u'v + uv'.
In this problem, the function v is a constant, denoted by c. Therefore, v(x) = c and its derivative v'(x) = 0, because the derivative of a constant is zero.
Substitute v = c and v' = 0 into the Product Rule formula: (u*c)' = u'c + uc'.
Since c' = 0, the formula simplifies to: (u*c)' = u'c.
This result aligns with the Derivative Constant Multiple Rule, which states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function: (c*u)' = c*u'.

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Derivative Product Rule

The Derivative Product Rule states that the derivative of the product of two functions is given by the formula: (uv)' = u'v + uv', where u and v are functions of a variable. This rule allows us to differentiate products of functions systematically, ensuring that both functions are accounted for in the differentiation process.
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The Product Rule

Constant Function

A constant function is a function that does not change its value regardless of the input. In the context of the question, if the function v has a constant value c, its derivative v' is zero. This simplifies the application of the Derivative Product Rule, as the term involving v' will vanish.
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Derivative Constant Multiple Rule

The Derivative Constant Multiple Rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. Mathematically, if f(x) is a function and c is a constant, then (cf(x))' = c f'(x). This rule highlights how constants affect differentiation and is particularly relevant when considering the implications of a constant function in the Product Rule.
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