2.5 - Applying the Power Rule

Introduction

The power rule is one of the most famous and useful tools in calculus. It’s a shortcut that lets you find derivatives quickly without using the limit definition.

Previously, to find the derivative of

f(x) = x^2

, we had to use:

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

That process was algebraically intensive. Now we have a shortcut.

Discovering the Pattern

If we used the limit definition to find derivatives of power functions, we’d get:

Original Function	Derivative
$f(x) = x^2$	$f'(x) = 2x$
$f(x) = x^3$	$f'(x) = 3x^2$
$f(x) = x^4$	$f'(x) = 4x^3$
$f(x) = x^5$	$f'(x) = 5x^4$

See the pattern? The exponent becomes the coefficient, and the new exponent is one less than the original.

Where the Pattern Comes From

Here’s the limit definition applied to

f(x) = x^3

. Expand

(x + h)^3

first:

(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h} = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h}

= \lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2

The same thing happens for any whole-number power

x^n

. When you expand

(x + h)^n

, the

x^n

cancels, the next term is always

nx^{n - 1}h

, and every term after that has at least

h^2

in it. After you divide by

h

, those leftover terms still have an

h

, so they go to

0

. All that’s left is

nx^{n - 1}

The Power Rule Formula

For any function of the form

f(x) = x^n

, where

n

is any real number:

\boxed{\frac{d}{dx}[x^n] = nx^{n-1}}

In words: 1. Bring the exponent down as a coefficient 2. Subtract 1 from the exponent

Alternative notation: if

y = x^n

, then

y' = nx^{n-1}

Basic Examples

Example 1: Large Positive Exponent

Find the derivative of

y = x^{37}

\frac{dy}{dx} = 37x^{36}

Process: the exponent 37 comes down as the coefficient, and the new exponent is

37 - 1 = 36

Example 2: Another Basic Power

Find

y'

y = x^9

y' = 9x^8

Negative Exponents

Example 3: Reciprocal Function

Find the derivative of

y = \frac{1}{x}

Step 1: Rewrite with a negative exponent

y = x^{-1}

Step 2: Apply the power rule

\frac{dy}{dx} = -1 \cdot x^{-1-1} = -1 \cdot x^{-2}

Step 3: Rewrite with a positive exponent

\frac{dy}{dx} = -\frac{1}{x^2}

Example 4: Higher Negative Power

Differentiate

y = \frac{1}{x^4}

Rewrite:

y = x^{-4}

Apply the power rule:

y' = -4x^{-5}

Final answer:

y' = -\frac{4}{x^5}

Fractional Exponents (Radicals)

Example 5: Square Root

Find the derivative of

y = \sqrt{x}

Step 1: Rewrite as a fractional exponent

y = x^{1/2}

Step 2: Apply the power rule

y' = \frac{1}{2}x^{-1/2}

Step 3: Rewrite as a radical

y' = \frac{1}{2x^{1/2}} = \frac{1}{2\sqrt{x}}

Example 6: Seventh Root

Differentiate

y = \sqrt[7]{x^3}

Step 1: Rewrite as a fractional exponent

y = x^{3/7}

Step 2: Apply the power rule

y' = \frac{3}{7}x^{3/7 - 1}

Step 3: Simplify the exponent

\frac{3}{7} - 1 = \frac{3}{7} - \frac{7}{7} = -\frac{4}{7}

y' = \frac{3}{7}x^{-4/7}

Step 4: Rewrite with a positive exponent

y' = \frac{3}{7x^{4/7}} = \frac{3}{7\sqrt[7]{x^4}}

Simplifying Before Differentiating

Sometimes you need to simplify the function before applying the power rule.

Example 7: Quotient of Powers

Find

f'(x)

and evaluate

f'(7)

for

f(x) = \frac{x}{\sqrt{x}}

Step 1: Rewrite with exponents

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

Step 2: Simplify using exponent rules

f(x) = x^{1 - 1/2} = x^{1/2}

Step 3: Apply the power rule

f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}

Step 4: Evaluate at

x = 7

f'(7) = \frac{1}{2\sqrt{7}}

Interpretation: this is the slope of the tangent line to

f(x)

x = 7

Example 8: Product Under a Radical

Find

f'(8)

for

f(x) = \sqrt[3]{x} \cdot x^3

Step 1: Rewrite with exponents

f(x) = x^{1/3} \cdot x^3

Step 2: Add exponents, since the bases are the same

f(x) = x^{1/3 + 3} = x^{1/3 + 9/3} = x^{10/3}

Step 3: Apply the power rule

f'(x) = \frac{10}{3}x^{7/3}

Step 4: Evaluate at

x = 8

f'(8) = \frac{10}{3} \cdot 8^{7/3}

Step 5: Simplify

8^{7/3}

8^{7/3} = (\sqrt[3]{8})^7 = 2^7 = 128

f'(8) = \frac{10}{3} \cdot 128 = \frac{1280}{3}

Pro tip: when evaluating at specific values, compute roots before powers whenever you can. -

\sqrt[3]{8} = 2

is easy -

2^7 = 128

is manageable - That’s way easier than computing

8^7

first

Parallel Tangent Lines

Two curves have parallel tangent lines at certain

x

-values if their derivatives (their slopes) are equal at those points.

The Concept

Parallel lines have equal slopes. So in calculus, if

f

and

g

have parallel tangent lines at

x = a

, then:

f'(a) = g'(a)

Example 9: Finding Points of Parallel Tangency

Find all

x

-values where

f(x) = x^4

and

g(x) = x^3

have parallel tangent lines.

Step 1: Find both derivatives

f'(x) = 4x^3 \qquad g'(x) = 3x^2

Step 2: Set the derivatives equal

4x^3 = 3x^2

Step 3: Move everything to one side

4x^3 - 3x^2 = 0

Step 4: Factor

x^2(4x - 3) = 0

Step 5: Solve using the zero product property

x^2 = 0 \quad \text{or} \quad 4x - 3 = 0

x = 0 \quad \text{or} \quad x = \frac{3}{4}

Answer: the tangent lines are parallel at

x = 0

and

x = \frac{3}{4}

Visual Understanding

At $x = 0$ : both functions have horizontal tangent lines, with slope 0.

At $x = \frac{3}{4}$ : both functions have tangent lines with the same positive slope.

Important: parallel tangent lines don’t mean the curves intersect at those points. It just means their slopes match.

Practice Problems

Problem 1

Find the derivative of

h(x) = x^{15}

Answer:

h'(x) = 15x^{14}

Problem 2

Differentiate

y = \frac{1}{x^3}

Answer:

y = x^{-3} \quad \Rightarrow \quad y' = -3x^{-4} = -\frac{3}{x^4}

Problem 3

Find

\frac{dy}{dx}

for

y = \sqrt[4]{x}

Answer:

y = x^{1/4} \quad \Rightarrow \quad \frac{dy}{dx} = \frac{1}{4}x^{-3/4} = \frac{1}{4\sqrt[4]{x^3}}

Problem 4

Find

f'(4)

f(x) = x\sqrt{x}

Answer:

f(x) = x \cdot x^{1/2} = x^{3/2}

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

f'(4) = \frac{3}{2}\sqrt{4} = \frac{3}{2}(2) = 3

Problem 5

Where do

f(x) = x^2

and

g(x) = x^4

have parallel tangent lines?

Answer:

f'(x) = 2x, \quad g'(x) = 4x^3

2x = 4x^3

2x - 4x^3 = 0

2x(1 - 2x^2) = 0

x = 0 \quad \text{or} \quad x = \pm\frac{1}{\sqrt{2}}

Common Mistakes to Avoid

Mistake 1: Forgetting to subtract 1

❌ Wrong:

\frac{d}{dx}[x^5] = 5x^5

✓ Correct:

\frac{d}{dx}[x^5] = 5x^4

Mistake 2: Subtracting from negative exponents incorrectly

❌ Wrong:

\frac{d}{dx}[x^{-3}] = -3x^{-2}

✓ Correct:

\frac{d}{dx}[x^{-3}] = -3x^{-4}

, because

-3 - 1 = -4

Mistake 3: Subtracting from fractions incorrectly

❌ Wrong:

\frac{d}{dx}[x^{2/5}] = \frac{2}{5}x^{1/5}

✓ Correct:

\frac{d}{dx}[x^{2/5}] = \frac{2}{5}x^{-3/5}

, because

\frac{2}{5} - 1 = \frac{2}{5} - \frac{5}{5} = -\frac{3}{5}

Mistake 4: Computing large powers inefficiently

When evaluating something like

16^{5/2}

❌ Wrong approach: compute

16^5

first, which is over a million

✓ Correct approach: compute

\sqrt{16} = 4

first, then

4^5 = 1024

Mistake 5: Using the power rule when the variable is in the exponent

❌ Wrong:

\frac{d}{dx}[2^x] = x \cdot 2^{x - 1}

✓ Correct: the power rule only works when the base is the variable and the exponent is a constant, like

x^5

. In

2^x

, the variable is in the exponent, so it’s an exponential function. You’ll learn its derivative in Topic 2.7.

Key Takeaways

Power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$ for any real number $n$

It works for positive, negative, and fractional exponents

Process: bring the exponent down, then subtract 1 from the exponent

Always rewrite radicals and fractions as exponents first

Simplify expressions before differentiating when you can

The derivative gives the slope of the tangent line at any point

Parallel tangent lines occur where the derivatives are equal

When evaluating at specific values, take roots before powers for easier calculation

Be careful with negative and fractional exponents when subtracting 1

The power rule is much faster than the limit definition

You’ll use this rule constantly throughout calculus

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