Basic Rules: The Power Rule, Constants, and Sums

Why these four rules come first

Most functions you’ll differentiate are built from simple pieces: terms added together, each term a constant times a power of x. The four rules in this guide handle exactly that structure. One of them — the power rule — does the actual differentiating; the other three let you break a complicated expression into simple pieces and handle each piece on its own. Master these and a function like 2x⁴ − 3x² + 6x − 8 stops being one scary problem and becomes four easy ones. Better yet, these four rules never go away: every harder technique you learn later sits on top of them.

Rule 1: The Power Rule

d/dx[xⁿ] = n·xⁿ⁻¹

The workhorse of differentiation, and happily one of the easiest rules to use: bring the exponent down in front, then knock the exponent down by one. That’s the whole move.

• x³ → 3x² — the 3 comes down, the exponent drops to 2.
• x⁷ → 7x⁶
• x → 1 — because x is really x¹: the 1 comes down, and x⁰ = 1 disappears.

A nice bonus: the rule works for any exponent, not just whole numbers. Negative exponents (x⁻¹ → −x⁻²) and fractions (x^(1/2) → ½x^(−1/2), which is the derivative of √x) follow the exact same two-step move. Once the rule feels automatic on x³, you already know how to handle them too.

Rule 2: The Constant Rule

d/dx[c] = 0

The derivative of any constant is zero. The reasoning is immediate once you remember what a derivative measures: rate of change. A constant doesn’t change — its graph is a horizontal line, and a horizontal line has slope zero everywhere.

This applies to every constant, including the famous ones: d/dx[7] = 0, d/dx[π] = 0, d/dx[−½] = 0. If a term contains no variable, its derivative is zero, no matter how impressive the term looks.

Rule 3: The Constant Multiple Rule

d/dx[c · f(x)] = c · f′(x)

A constant coefficient “rides along”: park it out front, differentiate the function part, then multiply back through. To differentiate 5x³, hold the 5, apply the power rule to x³ to get 3x², and reattach: 5 · 3x² = 15x².

The intuition: multiplying a function by 5 stretches its graph vertically by a factor of 5, which makes every slope on it 5 times steeper. The constant scales the function, so it scales the rate of change by exactly the same amount.

Rule 4: The Sum and Difference Rule

d/dx[f(x) ± g(x)] = f′(x) ± g′(x)

Differentiation distributes over addition and subtraction: the derivative of a sum is the sum of the derivatives, term by term, with each term keeping its sign. This is the rule that lets you read a polynomial left to right and differentiate as you go.

It works because rates of change add. If your savings account grows by $50/month and your investment account by $30/month, your total balance grows by $80/month — the rate of the whole is the sum of the rates of the parts.

How to apply them, step by step

Differentiate f(x) = 4x³ + 7x − 9:

1. Split at the plus and minus signs (sum/difference rule). Three separate problems: 4x³, 7x, and −9.
2. Park the coefficients (constant multiple rule). The 4 and the 7 wait outside while you differentiate x³ and x.
3. Differentiate each piece. Power rule: x³ → 3x² and x → 1. Constant rule: −9 → 0.
4. Reassemble. 4 · 3x² + 7 · 1 + 0 = 12x² + 7.

f′(x) = 12x² + 7

Common mistakes to avoid

• Differentiating famous constants as if they were variables. Because π and e are letters, students reflexively apply the power rule to terms like π³ and write 3π². But π³ is just a number (about 31.0), and the derivative of a number is 0. How to avoid it: before differentiating any term, ask “does this contain the variable?” No variable → derivative is zero.
• Making the constant disappear. Seeing 5x² and reasoning “the derivative of 5 is 0, so this becomes 0 · x²” — wiping out the whole term. The constant rule applies to terms that are only a constant; a constant times a function uses the constant multiple rule instead, and the 5 survives: d/dx[5x²] = 10x. How to avoid it: a coefficient never vanishes — it multiplies the derivative.
• Dropping a term or a sign partway through. With four or five terms in play, the most common error isn’t calculus at all — it’s losing a − or skipping a term while reassembling. How to avoid it: write one differentiated piece directly under each original term, keeping the signs aligned in a column, before combining anything.

Worked example, start to finish

Differentiate g(x) = 2x⁴ − 3x² + 6x − 8.

Check your work

Now try a few on paper, then verify every step — not just the final answer — with the derivative calculator. It names the rule it applies at each line, so if your answer disagrees, you can find the exact step where your work went a different way. Good starting points:

And if your first few attempts don’t match — that’s the process working, not failing. Finding the step where you diverged is how the rules move from memorized to understood.