CELE Engineering Mathematics — Differential CalculusMemory Anchors

Tags

• formula
• power rule
• differentiation

Topic

Derivative Rules

Concept

Power Rule: d/dx (x^n) = nx^(n-1)

Anchor Id

A1

Difficulty

easy

Memory Aid

Imagine a CONSTRUCTION FOREMAN (Engineer Reyes) who always 'brings the exponent DOWN as a multiplier, then reduces the floor number by one.' He's building a 5-story building (x^5). He shouts: 'FIVE workers on this floor — write it out front! Then knock the top floor off — now it's a 4-story building!' Result: 5x^4. He does this every single time, no exceptions.

Anchor Type

micro_story

Why It Works

The story maps perfectly to the two mechanical steps of the power rule: (1) multiply by n, (2) reduce exponent by 1. The construction context resonates with civil engineering reviewees.

Example Usage

For y = 3x^4: Reyes brings the 4 down → 3·4 = 12, then knocks one floor off → x^3. Answer: y' = 12x^3.

Recall Trigger

Think of Engineer Reyes shouting 'Bring it down, knock one off!'

Tags

• formula
• product rule
• differentiation

Topic

Derivative Rules

Concept

Product Rule: (uv)' = u'v + uv'

Anchor Id

A2

Difficulty

easy

Memory Aid

Remember: 'DASH-KEEP, KEEP-DASH' — Differentiate the first (DASH it), Keep the second (KEEP it); then Keep the first, Dash the second. Two terms, always added together. Chant it: 'Dash-Keep PLUS Keep-Dash!' Write it on your palm during the exam break.

Anchor Type

mnemonic

Why It Works

The two-word rhythm 'Dash-Keep / Keep-Dash' maps directly to the two terms of the product rule. The PLUS between them is built into the chant's pause.

Example Usage

y = x^2 · sin x → Dash x^2 = 2x, Keep sin x → first term: 2x·sin x. Keep x^2, Dash sin x = cos x → second term: x^2·cos x. Answer: y' = 2x sin x + x^2 cos x.

Recall Trigger

Hear 'Dash-Keep Plus Keep-Dash' in your head the moment you see two functions multiplied.

Tags

• formula
• quotient rule
• differentiation

Topic

Derivative Rules

Concept

Quotient Rule: (u/v)' = (u'v - uv') / v^2

Anchor Id

A3

Difficulty

medium

Memory Aid

Sing to the tune of 'Bahay Kubo': 'LOW-d-HIGH minus HIGH-d-LOW, all over the square of what's below!' Low = denominator (v), High = numerator (u), d = derivative. The minus sign is the key difference from the product rule — it's SUBTRACTION, not addition.

Anchor Type

rhyme

Why It Works

The Bahay Kubo tune is deeply familiar to every Filipino student, making the rhyme immediately catchy. 'Low' and 'High' are spatial memory cues for denominator and numerator.

Example Usage

y = x^2 / sin x. High = x^2, Low = sin x. d-High = 2x, d-Low = cos x. Numerator: (2x)(sin x) − (x^2)(cos x). Denominator: sin^2 x. Done.

Recall Trigger

Hum 'Bahay Kubo' and the words 'Low-d-High minus High-d-Low' automatically follow.

Tags

• formula
• chain rule
• composite functions

Topic

Derivative Rules

Concept

Chain Rule: d/dx f(g(x)) = f'(g(x)) · g'(x)

Anchor Id

A4

Difficulty

medium

Memory Aid

Think of it as a NESTED JEEPNEY ROUTE. To get to Quiapo (the outermost function), you first ride from your barangay to the terminal (inner function g), then transfer to the Quiapo jeep (outer function f). The chain rule says: 'Differentiate the outer jeep's rate × the inner jeep's rate.' You always multiply the two rates together — it's a chain of transfers.

Anchor Type

analogy

Why It Works

The jeepney transfer analogy perfectly captures 'composition of functions.' Filipino students relate immediately to connecting routes. The multiplicative chaining of rates mirrors the chain rule's multiplication.

Example Usage

y = (2x^2 + 1)^4. Outer function: ( )^4, inner: 2x^2+1. Outer derivative: 4( )^3 × inner derivative: 4x. Chain: y' = 4(2x^2+1)^3 · 4x = 16x(2x^2+1)^3.

Recall Trigger

Picture transferring jeepneys: outer rate × inner rate.

Tags

• definition
• critical point
• optimization

Topic

Maxima and Minima

Concept

Critical Point: f'(x) = 0 locates potential maxima and minima

Anchor Id

A5

Difficulty

easy

Memory Aid

Visualize a ROLLER COASTER (like Star City's ride). At the very TOP of a hill (maximum) the coaster is momentarily FLAT — slope = zero. At the very BOTTOM of a valley (minimum), also FLAT — slope = zero. The derivative measures slope. When the slope is zero, you are at a peak or a trough. The coaster doesn't stop forever — it's just MOMENTARILY flat.

Anchor Type

visual_association

Why It Works

The roller coaster peak/valley is a concrete, visceral image of slope = 0 at extrema. The 'momentarily flat' idea prevents confusion with horizontal asymptotes.

Example Usage

Find extrema of f(x) = x^3 − 3x. Set f'(x) = 3x^2 − 3 = 0 → x = ±1. These are the 'flat' points on the roller coaster.

Recall Trigger

See the roller coaster frozen at the top of its hill — slope is zero there.

Tags

• classification
• second derivative
• concavity

Topic

Maxima and Minima

Concept

Second Derivative Test: f'' > 0 → minimum (concave up), f'' < 0 → maximum (concave down)

Anchor Id

A6

Difficulty

easy

Memory Aid

Remember the SMILE vs FROWN test. A positive f'' means the curve smiles (concave UP, like a bowl holding water) → it's a MINIMUM — the lowest point of the smile. A negative f'' means the curve frowns (concave DOWN, like an upside-down bowl) → it's a MAXIMUM — the top of the frown. Positive = SMILE = MIN. Negative = FROWN = MAX.

Anchor Type

visual_association

Why It Works

The smile/frown association creates a direct visual-emotional link. 'Smile = happy = positive' and the shape of a smile is a valley (minimum). This prevents the common mix-up.

Example Usage

f(x) = −x^2 + 4. f''(x) = −2 < 0. The curve FROWNS → x = 0 is a maximum. f(0) = 4 is the max value.

Recall Trigger

Ask: 'Is the curve smiling or frowning at this point?'

Tags

• process
• optimization
• procedure

Topic

Maxima and Minima

Concept

Optimization Procedure: 5 steps to find maximum or minimum value

Anchor Id

A7

Difficulty

medium

Memory Aid

Use the acronym D-E-D-S-C: 'DEar Dear Sir/Contractor!' D = Define variables. E = Express the quantity to optimize in one variable (use the constraint). D = Differentiate and set to zero. S = Solve for the critical value. C = Confirm max or min using f'' (second derivative test) or check endpoints. Every optimization problem on the board exam follows DEDSC.

Anchor Type

acronym

Why It Works

The acronym DEDSC provides a linear checklist. The phrase 'Dear Sir/Contractor' (evoking a letter to a construction client) is memorable and contextually relevant to civil engineering practice.

Example Usage

Box problem: Define x = side cut. Express V = x(30-2x)^2. Differentiate V' = 0. Solve x = 5 cm. Confirm with V'' < 0 → maximum.

Recall Trigger

Hear 'Dear Sir/Contractor' → recall D-E-D-S-C steps.

Tags

• formula
• limits
• L'Hopital

Topic

Limits and Continuity

Concept

L'Hôpital's Rule for 0/0 or ∞/∞ indeterminate forms

Anchor Id

A8

Difficulty

medium

Memory Aid

Imagine L'Hôpital (pronounced 'Lo-pee-tal') as a hospital doctor (ospital = hospital in Filipino). When a math patient is SICK (indeterminate form 0/0 or ∞/∞), the doctor's prescription is: 'Differentiate the top, differentiate the bottom — separately! Don't use the quotient rule! Take their derivatives separately, then re-evaluate the limit.' The patient recovers (the limit now works). If still sick, treat again (apply again).

Anchor Type

micro_story

Why It Works

The ospital wordplay (Filipino for hospital) creates a memorable phonetic anchor. The 'sick patient' metaphor explains exactly when to apply the rule and what the common mistake is (not using quotient rule).

Example Usage

lim(x→0) sin x / x → 0/0 (sick!). Treat: d(sin x)/dx = cos x, d(x)/dx = 1. New limit: cos(0)/1 = 1.

Recall Trigger

Ospital / L'Hôpital — sick limit gets treated by separate differentiation.

Tags

• process
• related rates
• common mistake

Topic

Related Rates

Concept

Related Rates — Golden Rule: Substitute numbers ONLY AFTER differentiating

Anchor Id

A9

Difficulty

hard

Memory Aid

Remember: 'DIFF FIRST, PLUG LAST.' This is the most common board exam trap. If you plug in the given values before differentiating, the constants vanish and you get zero for every rate — the answer will be wrong. DIFF FIRST (differentiate the equation with respect to t), PLUG LAST (then substitute the known values).

Anchor Type

mnemonic

Why It Works

The two-word command is short, direct, and easy to recall under pressure. It directly addresses the most common student error on related rates problems.

Example Usage

Sphere: V = (4/3)πr^3. DIFF FIRST: dV/dt = 4πr^2 · dr/dt. PLUG LAST: 10 = 4π(2^2) · dr/dt → dr/dt = 0.199 m/s.

Recall Trigger

See any related rates problem → mentally shout 'DIFF FIRST, PLUG LAST!'

Tags

• formula
• radius of curvature
• geometry

Topic

Curvature

Concept

Radius of Curvature: R = [1 + (y')^2]^(3/2) / |y''|

Anchor Id

A10

Difficulty

hard

Memory Aid

Break the formula into three chunks: TOP = [1 + (y')^2]^(3/2). Think: '1 plus slope-squared, raised to the 3/2.' BOTTOM = |y''| (absolute value of second derivative). The structure is (1 + slope^2)^(3/2) over |curvature-rate|. Remember '3/2 on top, second deriv below.' The 3/2 power is distinctive — no other curvature formula in the exam uses 3/2.

Anchor Type

chunking

Why It Works

Chunking the formula into top and bottom reduces cognitive load. Labeling the top '1 + slope^2 raised to 3/2' and the bottom 'second derivative' creates two distinct memory slots.

Example Usage

y = x^2 at origin: y' = 0, y'' = 2. R = [1+0]^(3/2) / |2| = 1/2 = 0.5 units.

Recall Trigger

Three-halves power on top, second derivative below.

Tags

• formula
• trigonometry
• derivatives

Topic

Derivative Rules

Concept

Derivatives of trig functions: sin→cos, cos→−sin, tan→sec², cot→−csc², sec→sec·tan, csc→−csc·cot

Anchor Id

A11

Difficulty

medium

Memory Aid

For the co-functions (cosine, cotangent, cosecant): the derivative ALWAYS picks up a NEGATIVE sign. Remember: 'CO-functions are CO-negative.' sin→+cos (positive, no 'co' prefix on result), cos→−sin (negative! it's a co-function), tan→+sec², cot→−csc² (negative! co-function), sec→+sec·tan, csc→−csc·cot (negative! co-function). The pattern: co-functions → negative derivatives.

Anchor Type

mnemonic

Why It Works

The CO-negative rule covers 3 of the 6 trig derivatives with one pattern. Students no longer need to memorize all six separately — just know sine, tan, sec and negate for their co-counterparts.

Example Usage

Differentiate y = cos 3x. Since cosine is a co-function → negative. Apply chain rule: y' = −sin 3x · 3 = −3 sin 3x.

Recall Trigger

'CO-functions are CO-negative' → cosine, cotangent, cosecant all have negative signs.

Tags

• formula
• exponential
• logarithm

Topic

Derivative Rules

Concept

Derivative of e^x and ln x

Anchor Id

A12

Difficulty

easy

Memory Aid

e^x is the PERFECT NARCISSIST of calculus — its derivative is exactly itself: (e^x)' = e^x. It looks in the mirror and sees itself unchanged. ln x is the INVERSE PARTNER — its derivative is 1/x, which is surprisingly simple: (ln x)' = 1/x. Together they are the mirror couple: e^x → e^x (no change), ln x → 1/x (simplified drastically).

Anchor Type

analogy

Why It Works

The narcissist analogy for e^x is humorous and unique — students remember 'the function that differentiates to itself.' The mirror couple idea links the two related functions.

Example Usage

y = e^(3x): chain rule → y' = e^(3x) · 3 = 3e^(3x). y = ln(x^2+1): y' = 1/(x^2+1) · 2x = 2x/(x^2+1).

Recall Trigger

e^x is the narcissist — sees itself in the derivative mirror.

Tags

• formula
• tangent line
• geometry

Topic

Tangent Lines

Concept

Tangent Line Equation at a Point: y − y₀ = f'(x₀)(x − x₀)

Anchor Id

A13

Difficulty

easy

Memory Aid

Picture a SURVEYOR's LEVEL STAFF touching a curved road (like a Philippine highway). The staff is TANGENT to the road at one point. The slope of the staff is f'(x₀). To write the line equation: you know one point on the road (x₀, y₀) and you know the slope — just use point-slope form. The surveyor's staff always touches the curve at exactly ONE point without crossing.

Anchor Type

visual_association

Why It Works

The surveyor image is profession-specific and visual. The point-slope form connection gives students a bridge from their algebra background to the calculus application.

Example Usage

Find tangent to y = x^3 at (1,1). f'(x) = 3x^2, f'(1) = 3. Tangent: y − 1 = 3(x − 1) → y = 3x − 2.

Recall Trigger

Surveyor's staff touching the curve at one point — slope = f'(x₀).

Tags

• definition
• partial derivatives
• multivariable

Topic

Partial Derivatives

Concept

Partial Derivatives: differentiate w.r.t. one variable, treat all others as constants

Anchor Id

A14

Difficulty

medium

Memory Aid

Think of a MULTI-STORY PARKING BUILDING (like SM's parking). To study how traffic changes as you move along ONE floor (x direction), you FREEZE all other floors (hold y and z constant). You study movement in only one direction at a time. Partial derivative is just zooming into one direction while the rest of the world stands still — like pressing PAUSE on all other variables.

Anchor Type

analogy

Why It Works

The parking building analogy uses a multi-dimensional physical space that civil engineers understand. 'Freeze' and 'pause' are memorable action words for treating other variables as constants.

Example Usage

f(x,y) = 3x^2y + y^3. ∂f/∂x: treat y as frozen constant → 6xy. ∂f/∂y: treat x as frozen constant → 3x^2 + 3y^2.

Recall Trigger

Multi-story parking — freeze all floors except the one you're analyzing.

Tags

• process
• optimization
• constraint

Topic

Maxima and Minima

Concept

Optimization — Reducing to One Variable Using a Constraint

Anchor Id

A15

Difficulty

medium

Memory Aid

Engineer Santos has a budget constraint (like every Filipino construction project). He wants to maximize the area of a rectangular lot but only has 40 m of fencing. His project manager says: 'You can't have two unknowns — eliminate one using the budget (constraint)!' He writes x + y = 20, so y = 20 − x. Now area A = x(20−x) is in ONE variable. 'Use the constraint to fire one variable!' he says.

Anchor Type

micro_story

Why It Works

Budget constraints are a relatable Filipino context. The 'fire one variable' phrase is vivid and actionable. The story models the exact algebraic step needed.

Example Usage

Box problem: S = lw, constraint l + w = P/2. Substitute w = P/2 − l into area formula → single variable → differentiate.

Recall Trigger

Engineer Santos' budget — eliminate one variable with the constraint.

Tags

• process
• endpoint
• closed interval

Topic

Maxima and Minima

Concept

Endpoint Extrema — Checking Domain Boundaries

Anchor Id

A16

Difficulty

hard

Memory Aid

Always remember: 'CHECK THE FENCE.' When optimizing on a CLOSED interval [a, b], the absolute maximum or minimum might be at the boundary (the fence), not at the critical point inside. Students often find only the interior critical point and declare victory. But the extreme value could be at x = a or x = b. ALWAYS evaluate f at both critical points AND at the fences (a and b).

Anchor Type

mnemonic

Why It Works

The fence metaphor is spatial and easy to recall. It reminds students that a bounded domain has literal 'walls' that must be checked — a direct counterpart to the interior critical points.

Example Usage

Maximize f(x) = −x^2 + 4x on [0, 5]. Critical: f'= −2x+4 = 0 → x=2. Check: f(0)=0, f(2)=4, f(5)=−5. Max is 4 at x=2; min is −5 at the fence (x=5).

Recall Trigger

'Check the fence' — evaluate the function at both endpoints of the interval.

Tags

• process
• related rates
• Pythagorean

Topic

Related Rates

Concept

Ladder-Sliding Related Rates (Pythagorean setup)

Anchor Id

A17

Difficulty

hard

Memory Aid

Si Mang Jose ay nagtatayo ng hagdan (Mang Jose is setting up a ladder) against a wall. The ladder, the wall, and the floor form a RIGHT TRIANGLE. He knows: x^2 + y^2 = L^2 (Pythagorean theorem). When the base slides OUT, differentiate both sides with respect to t: 2x(dx/dt) + 2y(dy/dt) = 0 (L is constant!). Mang Jose remembers: '2x times base-rate plus 2y times wall-rate equals zero — they balance!'

Anchor Type

micro_story

Why It Works

The Mang Jose character is a relatable Filipino handyman. The ladder is a classic geometry setup. The 'balance' idea explains why the sum equals zero — conservation of the constant hypotenuse.

Example Usage

L=5m, x=3m, dx/dt=0.5m/s. First: y = √(25−9) = 4m. Then: 2(3)(0.5) + 2(4)(dy/dt) = 0 → dy/dt = −0.375 m/s (moving down).

Recall Trigger

Mang Jose's ladder — Pythagorean, differentiate, 2x·dx/dt + 2y·dy/dt = 0.

Tags

• definition
• limits
• continuity

Topic

Limits and Continuity

Concept

Limit Definition Concept: A limit describes approach, not arrival

Anchor Id

A18

Difficulty

easy

Memory Aid

Think of a TREN (train) approaching TUTUBAN station in Manila. The limit is what the train is HEADING TOWARD — even if the track has a gap (hole) exactly at Tutuban, the train still approaches from that direction. In calculus, f(a) might not exist (the station has a hole) but lim(x→a) f(x) still exists (the train still approaches). The limit is about the JOURNEY, not the ARRIVAL.

Anchor Type

analogy

Why It Works

The Tutuban train image is quintessentially Filipino and spatial. The 'gap in the track' maps perfectly to a removable discontinuity where f(a) is undefined but the limit exists.

Example Usage

lim(x→2) (x^2−4)/(x−2) → if x=2 the denominator is 0 (hole in track). Factor: (x−2)(x+2)/(x−2) = x+2. Limit = 4, even though f(2) is undefined.

Recall Trigger

Tren approaching Tutuban — limit is about approach, not arrival.

Tags

• definition
• increasing
• decreasing

Topic

Maxima and Minima

Concept

Increasing/Decreasing Functions: f'(x) > 0 increasing, f'(x) < 0 decreasing

Anchor Id

A19

Difficulty

easy

Memory Aid

Imagine driving UP Kennon Road (Baguio bound, POSITIVE slope → function increasing → f' > 0) versus driving DOWN from Baguio to Pangasinan (NEGATIVE slope → function decreasing → f' < 0). The derivative is literally the GRADE of the road at any point. Positive grade = climbing = increasing function. Negative grade = descending = decreasing function.

Anchor Type

visual_association

Why It Works

Kennon Road is a famous Philippine road with a clear uphill/downhill narrative. Civil engineers relate to road grades (slopes). The physical driving experience creates a kinesthetic memory.

Example Usage

f(x) = x^3 − 3x. f'(x) = 3x^2 − 3. f'>0 when x^2>1 → x>1 or x<−1 (driving uphill). f'<0 when −1<x<1 (driving downhill).

Recall Trigger

Driving up Kennon = positive derivative = increasing. Driving down = negative = decreasing.

Tags

• formula
• related rates
• sphere

Topic

Related Rates

Concept

Related Rates — Sphere Volume Formula Differentiated

Anchor Id

A20

Difficulty

medium

Memory Aid

MEMORIZE this chain: V = (4/3)πr³ → dV/dt = 4πr² · dr/dt. Chunk it: 'Four-thirds pi r-cubed BECOMES four pi r-squared.' The 4/3 multiplied by the exponent 3 gives exactly 4 — clean and beautiful! And 4πr² is just the SURFACE AREA of the sphere. So the rate of volume change = surface area × rate of radius change. Logical: the bigger the surface, the faster it fills.

Anchor Type

chunking

Why It Works

Noting that 4πr² = surface area creates a conceptual link that makes the formula memorable and logical. The multiplication 4/3 × 3 = 4 being 'clean' creates a satisfying pattern.

Example Usage

dV/dt = 10 m³/s at r = 2 m: 10 = 4π(4)(dr/dt) → dr/dt = 10/(16π) = 0.199 m/s.

Recall Trigger

Volume differentiated gives surface area times dr/dt — beautiful and logical!

Formula

d/dx (x^n) = nx^(n-1)

Mnemonic

ENGINEER REYES: Bring the exponent DOWN in front, then KNOCK one floor off. 'n down, then n minus one.'

When To Use

Any time you differentiate a power of x: polynomials, radicals (rewrite as fractional exponents), and reciprocals (rewrite as negative exponents).

What Each Part Means

n = the original exponent (multiplied to the front); x^(n-1) = the variable raised to one less than the original power

Formula

(uv)' = u'v + uv'

Mnemonic

DASH-KEEP + KEEP-DASH: Differentiate first, keep second → add → Keep first, differentiate second.

When To Use

Whenever two separate functions are MULTIPLIED together and you need to differentiate their product.

What Each Part Means

u' = derivative of first function; v = second function kept as is; u = first function kept; v' = derivative of second function

Formula

(u/v)' = (u'v − uv') / v²

Mnemonic

Bahay Kubo tune: LOW-d-HIGH minus HIGH-d-LOW, all over the square of what's below. LOW = denominator, HIGH = numerator, d = derivative.

When To Use

When one function is DIVIDED by another. Note the minus sign (unlike product rule's plus sign).

What Each Part Means

u' = derivative of numerator; v = denominator unchanged; u = numerator unchanged; v' = derivative of denominator; v² = denominator squared

Formula

d/dx f(g(x)) = f'(g(x)) · g'(x)

Mnemonic

NESTED JEEPNEY: Differentiate outer function (leave inner alone) TIMES differentiate inner function. Outer derivative × Inner derivative.

When To Use

Whenever a function is INSIDE another function — a composition. Recognizable by expressions like (...)^n, e^(expression), sin(expression), ln(expression).

What Each Part Means

f'(g(x)) = derivative of outer function evaluated at the inner function; g'(x) = derivative of the inner function

Formula

R = [1 + (y')²]^(3/2) / |y''|

Mnemonic

THREE-HALVES on top, SECOND DERIV below. Top: 1 + slope-squared, raised to 3/2. Bottom: absolute value of y double prime.

When To Use

Finding the radius of curvature of a curve y = f(x) at a specific point. Relevant in highway and railway alignment design (minimum radius of curves).

What Each Part Means

y' = first derivative (slope of curve); (y')² = slope squared; 1 + (y')² = base of the 3/2 power; y'' = second derivative (rate of change of slope); |y''| = always positive radius

Formula

dV/dt = 4πr² · dr/dt (sphere)

Mnemonic

Volume differentiated gives SURFACE AREA times dr/dt. Surface area of sphere = 4πr², so: rate of volume = surface area × rate of radius.

When To Use

Related rates problems involving spheres — balloon inflation, water droplet evaporation, spherical tank filling.

What Each Part Means

dV/dt = rate of change of volume; 4πr² = surface area of sphere; dr/dt = rate of change of radius with time

Formula

2x(dx/dt) + 2y(dy/dt) = 0 (ladder/Pythagorean related rates)

Mnemonic

Mang Jose's ladder: Differentiate x^2 + y^2 = L^2 with respect to t → 2x·x' + 2y·y' = 0. The 2s cancel later; the rates are opposite in sign because when base goes out, top goes down.

When To Use

Any Pythagorean right-triangle setup where two sides change with time (classic ladder problem, boat moving from dock, etc.).

What Each Part Means

dx/dt = rate base of ladder moves; dy/dt = rate top of ladder moves (negative = sliding down); L = constant ladder length (its derivative = 0)

Formula

lim f/g = lim f'/g' (L'Hôpital's Rule)

Mnemonic

Ospital treatment for sick limits (0/0 or ∞/∞): Differentiate top SEPARATELY, differentiate bottom SEPARATELY, then re-evaluate. NOT the quotient rule.

When To Use

ONLY when direct substitution gives 0/0 or ∞/∞ (indeterminate forms). Do NOT use when the limit evaluates to a normal fraction.

What Each Part Means

f = numerator function; g = denominator function; f' = derivative of numerator alone; g' = derivative of denominator alone

Chain Title

6 Trig Derivatives in Order

Recall Test

Cover the right side and recite: What is d/dx cos x? d/dx cot x? d/dx csc x? (All should be negative.)

Memory Chain

Story: 'SINA (sin) gave a COSINE (cos) to her friend. But COSa (cos) is negative — she SUBTRACTED a SIN(e). TANgina, SEC SQUARED (tan → sec²)! COT is NEGATIVE, CSC SQUARED (cot → −csc²). SEC is SEC-TAN (sec → sec·tan). CSC is NEGATIVE CSC-COT (csc → −csc·cot).' Pattern: CO-functions (cos, cot, csc) always get NEGATIVE signs.

Items To Remember

• d/dx sin x = cos x
• d/dx cos x = −sin x
• d/dx tan x = sec²x
• d/dx cot x = −csc²x
• d/dx sec x = sec x tan x
• d/dx csc x = −csc x cot x

Chain Title

DEDSC — 5 Steps of Optimization

Recall Test

What are the 5 steps of optimization? Write out DEDSC and explain each letter without looking.

Memory Chain

'Dear Engineer, Do Something Constructive!' D = Define. E = Express (eliminate variable). D = Differentiate (set to zero). S = Solve. C = Confirm. Five steps, five letters: DEDSC = Dear Engineer, Do Something Constructive.

Items To Remember

• Define variables clearly
• Express quantity to optimize in one variable using constraint
• Differentiate and set equal to zero
• Solve for critical value(s)
• Confirm maximum or minimum with second derivative test or endpoint check

Chain Title

Types of Indeterminate Forms for L'Hôpital

Recall Test

Name all 5 indeterminate forms. Which two can be directly treated with L'Hôpital without conversion?

Memory Chain

'ZERO OVER ZERO is the classic sick patient. INFINITY OVER INFINITY is the twin. Zero TIMES infinity is their cousin — convert to fraction to treat. Infinity MINUS infinity needs algebra surgery. The three power forms (0⁰, 1^∞, ∞⁰) need the logarithm injection before treatment.' Sick = indeterminate. Treatment = L'Hôpital after converting to fraction form.

Items To Remember

• 0/0
• ∞/∞
• 0 · ∞ (rewrite as fraction first)
• ∞ − ∞ (algebraic manipulation needed)
• 0⁰, 1^∞, ∞⁰ (take logarithm first)

Chain Title

Derivative Rules in Application Order

Recall Test

For each expression, identify which rule to use first: (a) x^5, (b) x^2·sin x, (c) x/(x+1), (d) sin(x^2), (e) x^2 + y^2 = 25.

Memory Chain

'PPQCI — Please Pass Quick Calculus Immediately!' P = Power. P = Product. Q = Quotient. C = Chain. I = Implicit. Each step is a new tool in the differentiation toolkit, used in this order of complexity.

Items To Remember

• 1. Power Rule (simplest — polynomials)
• 2. Product Rule (two functions multiplied)
• 3. Quotient Rule (two functions divided)
• 4. Chain Rule (composite/nested functions)
• 5. Implicit Differentiation (y not isolated)

Chain Title

Related Rates Problem-Solving Steps

Recall Test

Recite the 5 steps of related rates without looking. In which step do you substitute the given numerical values?

Memory Chain

'DRIDS — Draw, Relate, Implicit-differentiate, Data-substitute, Solve.' Say: 'DR. IDS' — like a doctor checking patient IDs. D = Draw diagram. R = Relate with a formula. I = Implicit diff w.r.t. t. D = substitute Data. S = Solve. DIFF FIRST, PLUG LAST is built into steps I and D.

Items To Remember

• 1. Draw and label a diagram
• 2. Identify the geometric/physical relationship (formula)
• 3. Differentiate BOTH SIDES with respect to t
• 4. Substitute known values and rates
• 5. Solve for the unknown rate