The power rule is one of the most useful shortcuts in calculus because it lets you differentiate powers of x quickly and reliably. Instead of using the limit definition every time, you can follow a simple pattern: bring the exponent down and subtract 1 from the exponent. This rule is the foundation for differentiating many polynomial, rational, and radical functions.
It matters because derivatives describe rates of change, slopes of curves, and motion in physics and engineering.
The rule says that if f(x) = x^n, then f'(x) = nx^(n - 1), where n can be a positive integer, negative integer, fraction, or even many real numbers. For a polynomial, you apply the power rule to each term separately and keep constant multipliers. Negative exponents help differentiate expressions like 1/x^2, while fractional exponents help differentiate roots like sqrt(x).
The power rule becomes even more powerful when combined with the constant multiple rule, sum rule, and chain rule.
Key Facts
- Power rule: if f(x) = x^n, then f'(x) = nx^(n - 1).
- Constant multiple rule: if f(x) = c x^n, then f'(x) = c n x^(n - 1).
- Sum rule: the derivative of a sum is the sum of the derivatives, so d/dx[f(x) + g(x)] = f'(x) + g'(x).
- Constant rule: if f(x) = c, then f'(x) = 0.
- Negative exponent example: d/dx(x^-3) = -3x^-4.
- Fractional exponent example: d/dx(x^(1/2)) = (1/2)x^(-1/2).
Vocabulary
- Derivative
- A derivative measures the instantaneous rate of change of a function or the slope of its graph at a point.
- Power Rule
- The power rule is the differentiation rule d/dx(x^n) = nx^(n - 1).
- Exponent
- An exponent tells how many powers of a base are being used, such as n in x^n.
- Coefficient
- A coefficient is a number multiplying a variable expression, such as 5 in 5x^3.
- Polynomial
- A polynomial is a sum of terms made from constants multiplied by nonnegative whole-number powers of a variable.
Common Mistakes to Avoid
- Forgetting to subtract 1 from the exponent: in d/dx(x^5), the derivative is 5x^4, not 5x^5, because the exponent must decrease by 1.
- Dropping the original coefficient: in d/dx(7x^3), the derivative is 21x^2, not 3x^2, because the coefficient 7 must be multiplied by the exponent.
- Treating constants like variables: in d/dx(9), the derivative is 0, not 9, because a constant has no change as x changes.
- Misusing negative or fractional exponents: in d/dx(x^-2), the derivative is -2x^-3, not 2x^-1, because the negative exponent still moves down and then decreases by 1.
Practice Questions
- 1 Find the derivative of f(x) = 4x^5 - 3x^2 + 7.
- 2 Find the derivative of g(x) = 6x^-2 + 8x^(1/2).
- 3 Explain why the power rule gives the slope of a horizontal line as 0 when the function is f(x) = 12.