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?

Verified step by step guidance

1. 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'.
2. 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.
3. Substitute v = c and v' = 0 into the Product Rule formula: (u*c)' = u'c + uc'.
4. Since c' = 0, the formula simplifies to: (u*c)' = u'c.
5. 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'.

Key Concepts

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.

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.

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.