Every formula, one page

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)?

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)?

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?

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?

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)?

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.

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?

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?

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?

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?

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?

Memory hook. Steady rate → straight line. Driving at 60 for x hours covers 60x miles.

On the graph. The area under a flat line of height k grows linearly: width × height = kx.

Watch out. Forgetting +C — and forgetting that ∫5 dx has an x in the answer.

Used for. Total cost from a fixed rate: hours × hourly rate.

Try it: What is ∫ 3 dx?

Memory hook. Reverse power rule: raise the power (1→2), divide by the new power (÷2).

On the graph. Area under y = x up to x is a triangle: ½ · base · height = x²/2.

Watch out. Forgetting to divide — ∫x dx is not x².

Used for. Distance under constant acceleration: speed grows like t, distance like t²/2.

Try it: What is ∫ x dx from 0 to 4? (Check with the triangle.)

Memory hook. Up one, divide by the new number. 2 → 3, then ÷3.

On the graph. The area under the parabola grows like a cubic — faster than the curve itself.

Watch out. Dividing by the old power (2) instead of the new one (3).

Used for. Volume by slicing: stacking square cross-sections of side x integrates x².

Try it: Differentiate x³/3 + 7 and confirm you get x² back.

Memory hook. The derivative power rule, rewound. The (n ≠ −1) fine print is where ln x lives.

On the graph. Each accumulation curve grows one power faster than its rate curve.

Watch out. Using it on x⁻¹ — that one's special, and the answer is ln|x|.

Used for. Any power-law rate — physics, economics, biology — accumulates by this one pattern.

Try it: What is ∫ x⁴ dx?

Memory hook. n = −1 would force division by zero, so nature invented ln to fill the gap.

On the graph. Accumulating 1/x grows without bound — but logarithmically slowly.

Watch out. Writing x⁰/0. Also: the absolute value matters — ln of a negative isn't real.

Used for. Time to grow between sizes under proportional growth involves ln of the ratio.

Try it: What is ∫ 1/x dx from 1 to e?

Memory hook. The selfie function works in both directions.

On the graph. The accumulated area under eˣ up to x is (almost) the height of eˣ itself.

Watch out. Forgetting +C — even self-similar functions shift.

Used for. Total growth from exponential rates: accumulated interest, accumulated population.

Try it: What is ∫ eˣ dx?

Memory hook. Derivative walks sin → cos forward; integral walks sin → −cos back.

On the graph. Area under one full hump of sine is exactly 2 — try it in the playground.

Watch out. Answering cos x. Differentiate your answer to check: (cos x)′ = −sin x. Wrong sign.

Used for. Net charge from an alternating current is the integral of its sine wave.

Try it: Differentiate −cos x and confirm you get sin x.

Memory hook. One backward step on the cycle: cos came from sin.

On the graph. Where cos is positive, the accumulated sine climbs; where negative, it falls.

Watch out. Adding a stray minus by analogy with ∫sin. Always verify by differentiating.

Used for. Position of an oscillator from its cosine-shaped velocity.

Try it: What is ∫ cos x dx from 0 to π/2?