Topic 2.5 Applying the Power Rule: differentiate power functions using the power rule, including negative and fractional exponents.

What this topic is asking

The College Board (Topic 2.5) introduces the power rule, the first shortcut that replaces the limit definition for differentiating powers of $x$. You must apply it to positive, negative, and fractional exponents, which means rewriting reciprocals and roots as powers first.

The rule

So $\frac{d}{dx}[x^7] = 7x^6$ and $\frac{d}{dx}[x^2] = 2x$, with no limit calculation needed.

Negative and fractional exponents

Why it works (briefly)

The power rule is not magic - it follows from the limit definition. Expanding $(x+h)^n$ with the binomial theorem, the difference quotient $\frac{(x+h)^n - x^n}{h}$ simplifies to $n x^{n-1}$ plus terms that vanish as $h \to 0$. The rule is the limit definition done once, in general, so you never have to repeat it.

The habit that prevents errors

Always rewrite to a single power first, differentiate, then translate back to roots and fractions if the question expects that form. Trying to differentiate $\frac{1}{x^2}$ or $\sqrt{x}$ in place is where most mistakes happen. With the rewrite habit, the power rule handles every algebraic power on the AP exam.

Splitting fractions and products into powers first

Many functions that do not look like a single power can be rewritten into a sum of powers, at which point the power rule applies term by term. A quotient with a single term in the denominator, such as $\frac{x^3 + 2x}{x}$, should be split into $\frac{x^3}{x} + \frac{2x}{x} = x^2 + 2$ before differentiating, which avoids needing the quotient rule entirely. Similarly, a product like $x^2(x + 3)$ is faster to expand to $x^3 + 3x^2$ and differentiate term by term than to invoke the product rule. The College Board deliberately includes questions where the slow path (quotient or product rule) and the fast path (rewrite, then power rule) both lead to the same answer, rewarding students who notice that algebraic simplification comes first. On the no-calculator section this judgement saves time and reduces the number of places an error can creep in.

What the power rule does not cover

The power rule applies to $x$ raised to a constant power, not to a constant raised to a variable power. So $\frac{d}{dx}[x^2] = 2x$ uses the power rule, but $\frac{d}{dx}[2^x]$ does not - that is an exponential function with its own rule. Likewise the power rule alone does not differentiate a power of a more complicated inside function, such as $(3x + 1)^5$; that requires the chain rule from Unit 3. Recognizing the boundary - constant exponent on a bare $x$ - keeps you from misapplying the rule to expressions that only superficially resemble a power. When in doubt, ask whether the base is exactly $x$ and the exponent is exactly a number; if so, the power rule is the tool.