The Power Rule: Differentiation Made Easy

📜 Formula of the Power Rule

For a function of the form $f(x) = x^n$ with a (real) exponent $n$, the derivative is:

💡 Good to know: This rule also works for negative and fractional exponents, as long as the expression is defined. For example, $x^{-1}$ and $x^{1/2}$ (i.e., $\sqrt{x}$) can be handled directly with the power rule.

🪜 Application in 3 Simple Steps

Use the power rule whenever you have a single power of the variable (possibly multiplied by a constant).

1. Identify the exponent: Write your term in the form $x^n$ (or $c \cdot x^n$ with constant $c$).
2. Bring down the exponent: Multiply by $n$ and reduce the exponent by one.
3. Simplify the result: If there is a constant factor, keep it. Combine like terms if needed.

✏️ Examples for Illustration

These examples show how the power rule applies to different kinds of exponents.

Example 1: Integer Exponent

Given the function: $f(x) = x^5$

• Apply the rule: $\dfrac{d}{dx}(x^5) = 5x^4$

Result: $f'(x) = 5x^4$

Example 2: Constant Factor

Given the function: $f(x) = 3x^4$

• Apply the rule: $\dfrac{d}{dx}(3x^4) = 3 \cdot 4x^3 = 12x^3$

Result: $f'(x) = 12x^3$

Example 3: Negative Exponent

Given the function: $f(x) = x^{-2}$

• Apply the rule: $\dfrac{d}{dx}(x^{-2}) = -2x^{-3} = -\dfrac{2}{x^3}$

Result: $f'(x) = -\dfrac{2}{x^3}$

Example 4: Fractional Exponent (Root)

Given the function: $f(x) = x^{1/2} = \sqrt{x}$ (for $x > 0$)

• Apply the rule: $\dfrac{d}{dx}\big(x^{1/2}\big) = \dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}$

Result: $f'(x) = \dfrac{1}{2\sqrt{x}}$

Example 5: $\boldsymbol{1/x}$ as a Power

Given the function: $f(x) = \dfrac{1}{x} = x^{-1}$ (for $x \neq 0$)

• Apply the rule: $\dfrac{d}{dx}(x^{-1}) = -1 \cdot x^{-2} = -\dfrac{1}{x^2}$

Result: $f'(x) = -\dfrac{1}{x^2}$

Example 6: Power of a Linear Function (Chain Rule hint)

Given the function: $f(x) = (2x+3)^4$

• Use power rule + chain rule: $$ \frac{d}{dx}\big((2x+3)^4\big) = 4(2x+3)^3 \cdot \frac{d}{dx}(2x+3) = 4(2x+3)^3 \cdot 2 = 8(2x+3)^3. $$

Result: $f'(x) = 8(2x+3)^3$