key term - Power Rule

Definition

The power rule is a fundamental mathematical concept that describes how to differentiate or integrate a function raised to a power. It is a crucial tool in calculus and algebra, allowing for the simplification and evaluation of expressions involving exponents.

5 Must Know Facts For Your Next Test

1. The power rule states that the derivative of $x^n$ is $nx^{n-1}$, and the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ plus a constant of integration.
2. The power rule is applicable when working with exponents in the context of properties of exponents, simplifying rational exponents, evaluating logarithmic functions, and using the properties of logarithms.
3. Rational exponents can be rewritten using the power rule, allowing for the simplification of expressions involving fractional exponents.
4. The power rule is used to differentiate and integrate logarithmic functions, as the logarithm function can be expressed using exponents.
5. The properties of logarithms, such as the power rule for logarithms, are closely related to the power rule for exponents and differentiation/integration.

Review Questions

• Explain how the power rule can be applied to simplify expressions involving exponents in the context of properties of exponents and scientific notation.
  • The power rule states that $x^n$ can be differentiated as $nx^{n-1}$ and integrated as $\frac{x^{n+1}}{n+1}$. This rule can be used to simplify expressions involving exponents, such as $x^3 \cdot x^4 = x^{3+4} = x^7$, or $\frac{x^5}{x^2} = x^{5-2} = x^3$. In the context of scientific notation, the power rule allows for the manipulation of exponents to convert between different forms of scientific notation.
• Describe how the power rule can be used to simplify rational exponents, and explain the connection to the properties of exponents.
  • The power rule can be applied to expressions with rational exponents, such as $x^{\frac{1}{2}}$ or $x^{\frac{3}{4}}$. By rewriting the rational exponent as a power, the power rule can be used to simplify the expression. For example, $x^{\frac{1}{2}} = \sqrt{x}$ and $x^{\frac{3}{4}} = (x^3)^{\frac{1}{4}}$. This connection between rational exponents and the power rule is derived from the properties of exponents, where $x^{\frac{a}{b}} = (x^a)^{\frac{1}{b}}$.
• Explain how the power rule is used in the context of evaluating and graphing logarithmic functions, and discuss the relationship between the power rule and the properties of logarithms.
  • The power rule is closely linked to the properties of logarithms, as the logarithm function can be expressed using exponents. When evaluating and graphing logarithmic functions, the power rule can be used to differentiate expressions like $\log_a(x^n) = n\log_a(x)$. This relationship between the power rule and logarithms is further demonstrated in the properties of logarithms, such as the power rule for logarithms: $\log_a(x^n) = n\log_a(x)$. Understanding the power rule is essential for manipulating and working with logarithmic functions and their properties.

Related terms

Exponent:

A number or variable that indicates how many times the base is used as a factor.

Rational Exponent:

An exponent that can be expressed as a ratio of two integers, such as $\frac{1}{2}$ or $\frac{3}{4}$.

Logarithm:

The inverse operation of exponentiation, used to express an exponent as a factor.