NAME THE TOOL

A square that grows

Coach

Remember Lesson 8? We proved the rate of x² is 2x, one h at a time. That was slow. Here is the shortcut.

A square with side x grows. Its area x² grows at 2x: the power 2 hops down in front, and the power drops by one, to x¹. Every power works this way.

Quick check: The power rule says the 2 in x² hops down in front. What is left as the power?

THE SYMBOLS

d/dx xⁿ = n · x^(n + (−1))

d/dx xⁿ = n · x^(n + (−1))

A number in front just rides along: d/dx 5x³ = 5 · 3x² = 15x². A constant alone has slope 0. For a sum, do each piece and add.

Quick check: d/dx x⁴?

WORKED EXAMPLE

f(x) = x⁴ + 2x. Find f′(1).

Strategy: Piece by piece

Name the rate or the pileTry a small caseDraw itAsk whyAnswer the exact thing asked

1

Understand

We want the slope at x = 1, not the height.

Why? f′ is a rate.

2

Plan

Power rule on each piece: x⁴ → 4x³; 2x → 2.

Why? Sums go term by term.

3

Solve

f′(x) = 4x³ + 2. At 1: 4 + 2 = 6.

Why? Plug in after differentiating.

4

Look back

Limit way for 2x gives 2; for x⁴ near 1 the secants head to 4. ✓

Why? Shortcut matches the slow way.

Picture way

The x⁴ part climbs at 4 there; the 2x part at 2. Total 6.

Symbol way

f′(1) = 4 · 1³ + 2 = 6

Quick check: True or false: f′(1) here is 3, because f(1) = 1 + 2.

SEE IT

Three ways to slip

Trap: Bringing the power down without lowering it

x⁵ → 5x⁵ keeps the old power. It is 5x⁴.

Trap: Lowering without bringing down

x⁵ → x⁴ forgets the 5 in front.

Trap: Keeping a constant

x² + 9 has slope 2x. The 9 is flat.

Quick check: d/dx (x³ + 8)?

WHAT TO LOOK FOR

Clue words and easy mix-ups

f′(a), slope at, rate atDifferentiate, then plug in.
f(a), value, heightPlug into f itself.
a plain number termSlope 0.
which function has slope …Run the power rule backward.

Quick check: Which is the slope of x² + 5x at x = 0?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

f(x) = x³ + 2. f′(3)? I plug in: 27 + 2 = 29.

Coach

That is f(3), the height. Differentiate first: 3x² + 0. At 3: 27.

Check habit: Differentiate first. Plug in second.

Quick check: f′(3) for x³ + 2?

TRAP CHECK

Run this before every answer

Power down in front.
Power lowered by 1.
Constants vanish; sums go piece by piece.
Only then plug in.