2.5 Applying the Power Rule

2.5 Applying the Power Rule

The power rule turns the derivative of a power function into a mechanical calculation: bring the exponent forward, then reduce the exponent by $1$. Together with the constant, sum, difference, and constant multiple rules, it transforms a complicated-looking polynomial into a short list of familiar derivatives.

The power rule

For a real exponent $r$, on the domain where the expression $x^r$ is defined and differentiable,

$$ \frac{d}{dx}\left(x^r\right)=rx^{r-1}. $$

The domain qualification matters. For example, $x^{1/2}$ is real-valued only when $x\ge 0$, and its derivative formula is applied on the interior of that domain, where the function is differentiable. Likewise, negative or fractional exponents may create excluded inputs or endpoint restrictions.

A useful visual pattern is:

$$ \boxed{x^r} \quad\longrightarrow\quad \boxed{r\cdot x^{r-1}} $$

The exponent moves into the coefficient slot; the exponent itself decreases by $1$.

For a polynomial term such as $7x^5$, first apply the power rule to $x^5$, then preserve the coefficient $7$:

$$ \frac{d}{dx}(7x^5) =7\frac{d}{dx}(x^5) =7(5x^4) =35x^4. $$

For a negative exponent,

$$ \frac{d}{dx}\left(x^{-3}\right) =-3x^{-4}. $$

For a fractional exponent,

$$ \frac{d}{dx}\left(x^{1/2}\right) =\frac12x^{-1/2}. $$

The rule changes the exponent; it does not multiply the entire expression by an unrelated quantity.

Rewrite first, then differentiate

Many expressions are powers in disguise. Rewrite roots and reciprocals with exponents before differentiating: $\sqrt[3]{x}=x^{1/3}$ has derivative $\frac13x^{-2/3}$, and $\frac{1}{x^2}=x^{-2}$ has derivative $-2x^{-3}$. The simplest case follows the same pattern: $\frac{d}{dx}(x)=1\cdot x^0=1$. The rule also agrees with the limit definition: for $x^2$, the difference quotient $\frac{(x+h)^2-x^2}{h}=2x+h$ approaches $2x$.

Misconception check

Misconception: “The derivative of $7x^4$ is $4x^3$.”

The exponent rule gives the derivative of $x^4$, not of the whole term. The coefficient must remain:

$$ \frac{d}{dx}(7x^4)=7(4x^3)=28x^3. $$

Retrieval check

This check is a full polynomial. Its terms are joined by the sum, difference and constant multiple rules of Topic 2.6, but every nonconstant term is a direct use of the power rule.

Differentiate

$$ q(x)=6x^4-2x^3+5x-14. $$

Apply the power rule, sum and difference rules, constant multiple rule, and constant-function rule:

$$ \boxed{q'(x)=24x^3-6x^2+5}. $$

If a coefficient or sign changes in your answer, identify the original term that caused it before continuing.