Topic 2.5 Notes – Applying the Power Rule

The Power Rule

You already know the derivative represents an instantaneous rate of change, defined using a limit. When that limit is applied to functions of the form $x^r$, it simplifies beautifully into a pattern:

$$\text{If } f(x) = x^r \text{ (where } r \text{ is a real constant), then } f'(x) = r x^{r-1}.$$

That’s it.

Two mechanical steps happen every time:

• Multiply by the exponent.
• Subtract 1 from the exponent.

The base $x$ stays the same.

Quick mental checks:

• $x^6 \rightarrow 6x^5$
• $x^2 \rightarrow 2x$
• $x \rightarrow 1$

This rule works for all real exponents: positive, negative, fractional, even zero.

Applying the Power Rule Correctly

Most mistakes happen before differentiation even starts. The key habit is rewriting expressions as powers of $x$.

Step-by-step process

1. Rewrite in exponent form
   • $\frac{1}{x^4} = x^{-4}$
   • $\sqrt{x} = x^{1/2}$
   • $\frac{1}{\sqrt{x}} = x^{-1/2}$
2. Apply $r x^{r-1}$ to each term
3. Simplify
   • Clean up coefficients.
   • Rewrite negative exponents if needed.

Negative exponents

Example:

$$f(x) = x^{-3}$$

Derivative:

$$f'(x) = -3x^{-4}$$

Often written as:

$$f'(x) = -\frac{3}{x^4}$$

Students frequently forget the negative sign from the exponent. It stays.

Fractional exponents

Example:

$$f(x) = x^{3/2}$$

Derivative:

$$f'(x) = \frac{3}{2}x^{1/2}$$

Which can be rewritten as:

$$\frac{3}{2}\sqrt{x}$$

If you see a radical, convert it first. It prevents mistakes.

Zero and first powers

• $x^0 = 1$, so derivative is $0$.
• $x^1 = x$, so derivative is $1$.

These show up constantly inside polynomials.

Polynomials and Constants

Polynomials are just sums of power functions.

If $$f(x) = 5x^4 - 3x^2 + 8x - 12$$

Differentiate term-by-term:

• $5x^4 \rightarrow 20x^3$
• $-3x^2 \rightarrow -6x$
• $8x \rightarrow 8$
• $-12 \rightarrow 0$

So:

$$f'(x) = 20x^3 - 6x + 8$$

Three reminders:

• Constants disappear.
• Coefficients stay and multiply.
• Each term is independent.

On a no-calculator MCQ, this is usually free points if you stay careful.

Seeing the Pattern Visually

Notice:

• The cubic has a flat slope at $x=0$.
• The derivative equals 0 there.
• The exponent dropped from 3 to 2.
• The new coefficient is 3.

This exponent drop happens every single time.

When This Rule Applies

Use the power rule immediately when:

• The function is $x^r$.
• You have a polynomial.
• Radicals or rational expressions can be rewritten as powers of $x$.

You are not expected to re-derive this from the limit definition unless explicitly asked. The AP exam treats this as a known rule.

Common Errors That Cost Points

• Forgetting to subtract 1 from the exponent.
• Dropping coefficients.
• Losing negative signs.
• Leaving answers unsimplified.
• Forgetting constants become 0.

Small algebra slips matter more than conceptual mistakes here.