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Lesson 06 · Formula fluency · 12 min

Basic derivative rules

You could compute every derivative from the limit definition — but you'd grow old doing it. A handful of patterns covers nearly every function you'll meet. Learn the pattern, and the formulas remember themselves.

Build the intuition

The power rule — one move, most functions

For any power of x: bring the power down in front, then lower the power by one. That single move handles x², x³, x¹⁰⁰, and — once you write them as powers — √x and 1/x too.

ddx xn=n x n−1\frac{d}{dx}\,x^n = n\,x^{\,n-1}

Constants vanish, sums split

Constants don't change, so their derivative is 0. Constant multipliers ride along unchanged: (5x²)′ = 5 · 2x. And the derivative of a sum is the sum of derivatives — differentiate term by term, like reading left to right.

The special characters

Three functions have derivatives worth knowing by heart: eˣ is its own derivative (that's its defining superpower), ln x has derivative 1/x (covering the one power the power rule can't), and sin/cos chase each other in a four-step cycle: sin → cos → −sin → −cos → back.

A taste of the chain rule

What about sin(x²) — a function inside a function? Differentiate the outside, then multiply by the derivative of the inside. Like gears: the outer wheel's speed times the inner wheel's speed. That's the chain rule, and it's a story for the next level — just know it exists.

See it move

InteractiveA function and its derivative

The function

Its derivative — the slope, graphed

1.3
Slope of the top curve at x = 1.3 is 0.29 — and that is exactly the height of the bottom curve there.

sin's slope, plotted point by point, traces out cos. The rules in the cards below are all visible facts like this one.

A worked example

Differentiate a polynomial in one pass

  1. Take

    f(x)=x3+5x2−7x+4f(x) = x^3 + 5x^2 - 7x + 4

  2. Term by term — power rule, constants riding along:

    f′(x)=3x2+10x−7f'(x) = 3x^2 + 10x - 7

  3. The +4 vanished (constants don't change). Each x lost one power. Done in one line.

The derivative cards — tap any to open

ddx(c)=0\frac{d}{dx}(c) = 0

A constant never changes, so its rate of change is zero.

Memory hook. Things that don't move have no speed. Constants vanish under the derivative.

On the graph. A flat horizontal line — the slope is 0 everywhere, by sight.

Watch out. Writing d/dx(5) = 5. The 5 isn't changing — the answer is 0.

Used for. Fixed costs in business: they don't affect the marginal (per-unit) cost.

Try it: What is d/dx(−12)?

ddx(x)=1\frac{d}{dx}(x) = 1

y = x rises one unit for every unit you move right — a steady rate of 1.

Memory hook. The power rule with n = 1: bring down the 1, lower the power to 0, and x⁰ = 1.

On the graph. A 45° line. Same slope (1) at every single point.

Watch out. Answering x or 0. The line is changing — at a constant rate of exactly 1.

Used for. Anything that grows one-for-one: distance at 1 m/s, cost at $1 per item.

Try it: What is d/dx(x + 7)?

ddx(x2)=2x\frac{d}{dx}(x^2) = 2x

The parabola steepens as x grows — its slope at any point is twice the x-value.

Memory hook. Power rule: bring the 2 down front, drop the power by one. 2 · x¹ = 2x.

On the graph. Flat at the bottom (slope 0 at x = 0), steeper and steeper as you move away.

Watch out. Forgetting the slope is negative on the left side — 2x is negative when x is.

Used for. Falling objects: height involves t², so speed involves 2t — speed grows over time.

Try it: What is the slope of x² at x = 3?

ddx(x3)=3x2\frac{d}{dx}(x^3) = 3x^2

The cubic's slope is 3x² — always positive (except a flat instant at 0).

Memory hook. Same dance: 3 steps down front, power drops to 2.

On the graph. Always climbing, with one flat moment at the origin where 3x² = 0.

Watch out. Writing 3x³ — you must reduce the power after bringing it down.

Used for. Volume of a cube is s³, so volume grows at 3s² — exactly its surface-to-edge sensitivity.

Try it: What is d/dx(x³) at x = −2?

ddx(xn)=n x n−1\frac{d}{dx}(x^n) = n\,x^{\,n-1}

The master pattern: every power follows the same two-step move.

Memory hook. Bring down the power, reduce by one. Five words that earn most of a calculus exam.

On the graph. Higher powers hug the axis near 0 and explode faster far from it.

Watch out. Applying it to eˣ or 2ˣ — the power rule needs the variable in the base, not the exponent.

Used for. Scaling laws everywhere: drag ∝ v², so its sensitivity to speed is 2v.

Try it: Use the pattern: d/dx(x¹⁰⁰)?

ddx(x)=12x\frac{d}{dx}\left(\sqrt{x}\right) = \frac{1}{2\sqrt{x}}

Square-root growth slows down — the bigger x gets, the gentler the climb.

Memory hook. √x is x^½. Power rule: ½ x^(−½) = 1/(2√x). New skin, same rule.

On the graph. Very steep near 0, then flattening forever — diminishing returns drawn as a curve.

Watch out. Guessing 1/√x — don't lose the ½ that comes down from the exponent.

Used for. Diminishing returns: doubling effort doesn't double results when output grows like √x.

Try it: Write √x as a power and differentiate it yourself.

ddx(1x)=−1x2\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}

1/x always decreases — so its rate of change is always negative.

Memory hook. 1/x is x⁻¹. Power rule: −1 · x⁻² = −1/x². The minus sign is the power coming down.

On the graph. Both branches slope downhill everywhere; the derivative is negative on both sides.

Watch out. Dropping the minus sign. 1/x is shrinking — the sign is the whole story.

Used for. Intensity fading with distance: how fast brightness drops as you step away.

Try it: What is d/dx(1/x) at x = 2?

ddx(ex)=ex\frac{d}{dx}\left(e^x\right) = e^x

eˣ grows at a rate equal to its own current size. It is its own derivative.

Memory hook. The selfie function — its derivative is a picture of itself. That's literally why e exists.

On the graph. At every point, the slope of eˣ equals the height of eˣ. One curve, two jobs.

Watch out. Applying the power rule to get x·e^(x−1). The variable is in the exponent — different rule.

Used for. Compound interest, population growth, epidemics: growth proportional to current amount.

Try it: If a quantity equals eˣ, how fast is it growing when it reaches 100?

ddx(ln⁡x)=1x\frac{d}{dx}(\ln x) = \frac{1}{x}

ln x keeps rising but ever more slowly — at rate exactly 1/x.

Memory hook. ln and 1/x are partners: the one power the power rule can't reach (x⁻¹) is covered by ln.

On the graph. Steep near 0, nearly flat far out. The slope at x = 100 is a tiny 1/100.

Watch out. Saying ln x flattens out to a ceiling. It never stops growing — just very slowly.

Used for. Perceived loudness and pitch are logarithmic — sensitivity falls as intensity rises.

Try it: What is the slope of ln x at x = 1?

ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x

The slope of the sine wave at any point is given by the cosine wave.

Memory hook. The cycle: sin → cos → −sin → −cos → sin. Differentiating walks one step around.

On the graph. Where sin peaks, cos crosses zero — flat tops have zero slope. It checks out visually.

Watch out. Writing −cos x. The minus appears when you differentiate cos, not sin.

Used for. A swinging pendulum: position is sine-like, so velocity is cosine-like.

Try it: What is the slope of sin x at x = 0?

ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x

Cosine starts at its peak — so it begins by falling, and its slope is minus sine.

Memory hook. Same four-step cycle, one step further along. The minus shows up on this step.

On the graph. At x = 0, cos is at the top of the hill: slope 0, then downhill — matching −sin.

Watch out. Forgetting the minus — the most common trig-derivative slip there is.

Used for. AC electricity: voltage and current are shifted waves of each other.

Try it: What is d/dx(cos x) at x = π/2?

Out in the world

Physics homework, solved by pattern

A ball's height is h(t) = 20t − 5t². Velocity is the derivative: h′(t) = 20 − 10t — power rule, twice. It hits zero at t = 2: the ball's peak, found by pattern-matching instead of head-scratching.

Common confusion, cleared

“The power rule works on eˣ and 2ˣ.”

Only when the variable is in the base (x³). With the variable in the exponent, the rules change — eˣ is its own derivative.

“d/dx(cos x) = sin x.”

It's −sin x. Cosine starts at its peak and immediately falls, so its slope starts negative. Picture the graph and the sign takes care of itself.

Check yourself

PracticeQuick check

  1. Match the derivative:

    ddx x5=  ?\frac{d}{dx}\,x^5 = \;?

  2. Which function is its own derivative?

  3. Match the derivative:

    ddx(1x)=  ?\frac{d}{dx}\left(\frac{1}{x}\right) = \;?

Recap

  • Power rule: bring down the power, reduce by one.
  • Constants vanish; sums split term by term.
  • eˣ is its own derivative; (ln x)′ = 1/x; sin → cos → −sin → −cos.