key term - Constant Multiple Rule

Definition

The constant multiple rule is a fundamental concept in mathematics that states that when a function is multiplied by a constant, the derivative of that function is equal to the constant multiplied by the derivative of the original function. This rule applies across various mathematical domains, including series, limits, and derivatives.

5 Must Know Facts For Your Next Test

1. The constant multiple rule states that if $f(x)$ is a function and $k$ is a constant, then $\frac{d}{dx}(kf(x)) = k\frac{d}{dx}f(x)$.
2. The constant multiple rule is particularly useful when finding the derivative of a function that is the product of a constant and another function.
3. In the context of series, the constant multiple rule can be used to simplify the evaluation of a series by factoring out a constant from the terms of the series.
4. When finding limits, the constant multiple rule can be used to simplify the limit of a function that is the product of a constant and another function.
5. The constant multiple rule is a fundamental property of derivatives and is often used in combination with other derivative rules, such as the power rule and the sum rule.

Review Questions

• Explain how the constant multiple rule can be applied to finding the derivative of a function.
  • The constant multiple rule states that if $f(x)$ is a function and $k$ is a constant, then $\frac{d}{dx}(kf(x)) = k\frac{d}{dx}f(x)$. This means that when a function is multiplied by a constant, the derivative of the resulting function is equal to the constant multiplied by the derivative of the original function. This rule is particularly useful when finding the derivative of a function that is the product of a constant and another function, as it allows you to simplify the derivative calculation.
• Describe how the constant multiple rule can be applied to the context of series and their notations.
  • In the context of series, the constant multiple rule can be used to simplify the evaluation of a series by factoring out a constant from the terms of the series. If a series is of the form $\sum_{n=1}^{\infty} kf(n)$, where $k$ is a constant and $f(n)$ is a function of the index $n$, then the constant multiple rule allows us to write the series as $k\sum_{n=1}^{\infty} f(n)$. This can greatly simplify the process of evaluating the series, as the constant factor can be pulled out of the summation.
• Analyze how the constant multiple rule can be used in the context of finding limits and their properties.
  • When finding limits, the constant multiple rule can be used to simplify the limit of a function that is the product of a constant and another function. If the limit of a function $f(x)$ is known, and the function $g(x)$ is a constant multiple of $f(x)$, then the constant multiple rule allows us to conclude that the limit of $g(x)$ is the constant multiple of the limit of $f(x)$. This property of limits is particularly useful when working with functions that can be expressed as the product of a constant and another function, as it allows us to leverage our knowledge of the limit of the other function to find the limit of the given function.

Related terms

Derivative:

The derivative of a function represents the rate of change of the function at a particular point, and is a fundamental concept in calculus.

Limit:

The limit of a function is the value that the function approaches as the input approaches a particular point, and is a crucial concept in finding the derivative of a function.

Series: A series is the sum of an infinite sequence of terms, and the constant multiple rule can be applied to the terms of a series to simplify its evaluation.