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Calculus mnemonics that make derivatives stick

Derivatives, integrals, and limits — the rules you need, in a form you'll actually remember.

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🔢 Calculus

Memory tricks

Proven mnemonics — fast to learn, hard to forget.

🔢 Calculus
Low d-High minus High d-Low, over Low Low
Quotient Rule
The quotient rule — never mix up the order again
For d/dx [f/g]: bottom times derivative of top, minus top times derivative of bottom, all over bottom squared. Say it out loud twice and you own it.
Difficulty: Intermediate
🔢 Calculus
"Bring it down, reduce by one"
Power Rule
Derivative of xⁿ — the most-used rule in calculus
d/dx [xⁿ] = n·xⁿ⁻¹. Bring the exponent down as a coefficient, then subtract 1 from the exponent. Fast, every time.
Difficulty: Beginner
🔢 Calculus
LIATE
Integration by Parts — which to pick as u
Choose u correctly for integration by parts every time
When integrating by parts (∫u dv), pick u in this priority order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
L
Logarithmic — ln(x), log(x)
I
Inverse trig — arcsin, arctan
A
Algebraic — x², x, polynomials
T
Trigonometric — sin, cos, tan
E
Exponential — eˣ, aˣ
Difficulty: Advanced
🔢 Calculus
"Derivative of sin is cos, cos is negative sin"
Trig Derivatives
Trig derivatives cycle — know the pattern
sin → cos → −sin → −cos → sin. The derivatives cycle every 4 steps. If you know sin and cos, you can derive all other trig derivatives.
Difficulty: Intermediate
🔢 Calculus
FTC: Area = F(b) − F(a)
Fundamental Theorem of Calculus
The FTC connects derivatives and integrals
To evaluate ∫[a to b] f(x) dx, find the antiderivative F(x), then compute F(b) − F(a). Plug in top, plug in bottom, subtract.
Difficulty: Intermediate
Chain Rule
Chain rule: derivative of f(g(x)) = f'(g(x)) · g'(x) — outside × inside'
Chain Rule
Differentiating composite functions — work from outside in
d/dx[sin(x²)] = cos(x²) · 2x. Outer function derivative × inner function derivative. Think: 'outside prime, keep inside, times inside prime.' Most common mistake: forgetting to multiply by the derivative of the inside.
Difficulty: Intermediate
Product Rule
Product rule: (uv)' = u'v + uv'
Product Rule
Differentiating the product of two functions
d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x). Memory trick: 'first times derivative of second, plus second times derivative of first.' Or: 'd(uv) = v du + u dv.' Never just multiply the derivatives — that's wrong.
Difficulty: Intermediate
Understanding Limits
Limit definition: lim as x→a means what value does f(x) approach as x gets close to a
Understanding Limits
The foundation of all calculus — approach, not arrive
The limit asks: what is f(x) heading toward as x approaches a? The function doesn't need to be defined AT x=a. lim x→2 of (x²-4)/(x-2): direct substitution gives 0/0. Factor: (x-2)(x+2)/(x-2) = x+2. Limit = 4.
Difficulty: Intermediate
Concavity and Inflection Points
Concave up (∪) = f''(x) > 0. Concave down (∩) = f''(x) < 0. Inflection = f''(x) = 0.
Concavity and Inflection Points
The second derivative tells you the shape of the curve
First derivative = slope (increasing/decreasing). Second derivative = curvature (concave up or down). Inflection point: where concavity changes (f''(x) = 0 AND sign changes). Concave up looks like a cup — holds water. Concave down — spills water.
Difficulty: Intermediate
Optimization
Optimization: find critical points (f'(x)=0), test with second derivative — positive = min, negative = max
Optimization
Finding maximum and minimum values using derivatives
Set f'(x) = 0 to find critical points. Second derivative test: f''(x) > 0 at critical point → local minimum. f''(x) < 0 → local maximum. Also check endpoints for absolute max/min on closed intervals. Real-world applications: maximize profit, minimize cost.
Difficulty: Advanced
U-Substitution
Integration by substitution: let u = inside function, du = inside derivative · dx
U-Substitution
Reversing the chain rule — the most common integration technique
∫2x·cos(x²)dx: let u = x², du = 2x dx. Substitute: ∫cos(u)du = sin(u) + C = sin(x²) + C. Step 1: choose u (usually the inside of a composite function). Step 2: find du. Step 3: substitute everything. Step 4: integrate. Step 5: back-substitute.
Difficulty: Advanced
Area Between Curves
Area between curves = ∫[top function − bottom function] dx from a to b
Area Between Curves
Finding the area sandwiched between two functions
Always subtract the lower function from the upper. Find intersection points first (set equal, solve for x) — these are your limits of integration a and b. If curves switch (one becomes the other's top), split the integral at the crossing point.
Difficulty: Advanced