Power rule

Definition

The power rule is a basic differentiation rule used to find the derivative of a function of the form $f(x) = x^n$. It states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

5 Must Know Facts For Your Next Test

1. The power rule applies to any real number exponent $n$, including negative and fractional exponents.
2. To use the power rule, multiply the original exponent by the coefficient and then subtract one from the exponent.
3. For constants (i.e., when $n=0$), the derivative of a constant is zero.
4. The power rule can be combined with other differentiation rules such as the product rule and chain rule.
5. In applications, the power rule helps simplify finding tangents, rates of change, and optimizing functions.

Review Questions

• What is the derivative of $f(x) = x^5$ using the power rule?
• How does the power rule apply if you have a function like $g(x) = \frac{1}{x^3}$?
• If $h(x) = \sqrt{x}$, how would you use the power rule to find $h'(x)$?

Related terms

Product Rule: A differentiation rule used when differentiating products of two functions. It is defined as $(fg)' = f'g + fg'$.

Chain Rule: A differentiation rule used for composite functions. If a function $y$ depends on $u$, which itself depends on $x$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

Quotient Rule: $A$ differentiation rule used for dividing two functions. It is given by $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$.