key term - Power Rule

Definition

The power rule is a fundamental concept in calculus that describes how to differentiate functions that involve exponents. It states that the derivative of a function raised to a power is equal to the power multiplied by the function raised to the power minus one, multiplied by the derivative of the function.

5 Must Know Facts For Your Next Test

1. The power rule is used to differentiate functions that involve exponents, such as $x^n$, where $n$ is any real number.
2. The power rule states that $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is the exponent and $x$ is the variable.
3. The power rule is a special case of the more general product rule, which is used to differentiate products of functions.
4. The power rule is particularly useful in simplifying the differentiation of polynomial functions, which are functions that are the sum of terms involving variables raised to non-negative integer powers.
5. The power rule can be extended to differentiate functions with variable exponents, such as $x^{f(x)}$, using the chain rule.

Review Questions

• Explain how the power rule is used to differentiate functions involving exponents.
  • The power rule states that the derivative of a function raised to a power is equal to the power multiplied by the function raised to the power minus one, multiplied by the derivative of the function. For example, if we have the function $f(x) = x^n$, the power rule tells us that the derivative of this function is $\frac{d}{dx}(x^n) = nx^{n-1}$. This allows us to quickly and easily differentiate functions that involve exponents, which is a common task in calculus.
• Describe how the power rule is related to the concept of integer exponents and scientific notation.
  • The power rule is closely related to the concepts of integer exponents and scientific notation, as these topics all involve manipulating functions that include exponents. In the context of integer exponents and scientific notation, the power rule can be used to differentiate expressions that involve numbers raised to integer powers, such as $10^x$ or $x^2$. Understanding the power rule is crucial for being able to work with and differentiate these types of functions, which are commonly encountered in pre-algebra and algebra courses.
• Analyze how the power rule can be extended to differentiate functions with variable exponents, and explain the importance of this extension.
  • The power rule can be extended to differentiate functions with variable exponents, such as $x^{f(x)}$, using the chain rule. This extension is important because it allows for the differentiation of a wider range of functions that involve exponents, including those where the exponent itself is a function of the variable. Being able to differentiate these types of functions is crucial in more advanced calculus topics, as they often arise in the study of optimization problems, differential equations, and other areas of mathematics. The ability to apply the power rule, even in the context of variable exponents, demonstrates a deep understanding of the underlying principles of differentiation.

Related terms

Derivative: A derivative is a measure of how a function changes as its input changes. It represents the rate of change of a function at a particular point.

Exponent:

An exponent is a number that indicates how many times the base number is multiplied by itself. It represents the power to which a number is raised.

Function:

A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)).