It's not you—it's the teaching.
Most calculus courses prioritize:
Goal of this guide: Build genuine understanding, not just exam performance.
Calculus in one sentence: "Calculus is the mathematics of change and accumulation."
The two big ideas:
1. Derivatives (Rate of Change) How fast is something changing right now?
You MUST be solid on:
Spend 2 weeks reviewing BEFORE calc starts. It's worth it.
What is a limit?
Informal: What value does f(x) approach as x gets closer to some number?
Example: What is lim(x→2) of (x² - 4)/(x - 2)?
Direct substitution gives 0/0 (indeterminate).
Factor: (x-2)(x+2)/(x-2) = x+2
As x→2, this approaches 4.
Why limits matter:
Graph f(x) = 1/x
What happens as x→0?
What happens as x→∞?
1. Direct Substitution (easiest) If f(x) is continuous at a, then lim(x→a) f(x) = f(a).
2. Factoring (for 0/0 forms) Example: lim(x→3) (x²-9)/(x-3) Factor numerator: (x-3)(x+3)/(x-3) = x+3 = 6
3. Conjugate Multiplication (for radicals) Example: lim(x→0) (√(x+1) - 1)/x Multiply by (√(x+1) + 1)/(√(x+1) + 1) Simplifies to 1/2
4. L'Hôpital's Rule (for 0/0 or ∞/∞) If lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g' (Take derivative of top and bottom separately)
Don't just do problems—understand WHY.
After solving, ask:
What is a derivative?
Geometric: Slope of tangent line at a point Physical: Instantaneous rate of change
Definition: f'(x) = lim(h→0) [f(x+h) - f(x)]/h
Example: Position vs. Time
s(t) = position at time t v(t) = s'(t) = velocity (how fast position changes) a(t) = v'(t) = s''(t) = acceleration (how fast velocity changes)
If s(t) = t²:
Power Rule: d/dx [x^n] = n·x^(n-1)
Why? x² = x·x, rate of change involves TWO factors changing.
Constant Multiple: d/dx [c·f(x)] = c·f'(x)
Why? Scaling doesn't change rate of change proportionally.
Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
Why? Rates of change add (like combining velocities).
Product Rule: d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
NOT f'(x)·g'(x)! (Common mistake)
Quotient Rule: d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²
Chain Rule (MOST IMPORTANT): d/dx [f(g(x))] = f'(g(x)) · g'(x)
Example: d/dx [sin(x²)] = cos(x²) · 2x
Why chain rule is hard: You're taking derivative of OUTER function, then multiplying by derivative of INNER function.
Trick: "Outside-inside" - derivative of outside (leaving inside alone), times derivative of inside.
d/dx [sin x] = cos x d/dx [cos x] = -sin x d/dx [tan x] = sec² x d/dx [e^x] = e^x d/dx [ln x] = 1/x d/dx [a^x] = a^x · ln a
Problem: You have 100 meters of fence. What dimensions maximize the area of a rectangular enclosure?
Setup: Let x = width, y = length Constraint: 2x + 2y = 100 → y = 50 - x Area: A(x) = x·y = x(50-x) = 50x - x²
Maximize A(x): A'(x) = 50 - 2x Set A'(x) = 0: 50 - 2x = 0 → x = 25
Check: A''(x) = -2 < 0, so x=25 is a maximum.
Answer: 25m × 25m (a square).
Key insight: Many real-world problems reduce to finding max/min of functions.
Problem: A ladder 10m long leans against a wall. The bottom slides away at 1 m/s. How fast is the top sliding down when the bottom is 6m from the wall?
Setup: x = distance from wall (bottom) y = height on wall (top) Pythagorean: x² + y² = 100
Differentiate with respect to time: 2x(dx/dt) + 2y(dy/dt) = 0
Given: dx/dt = 1, x = 6 Find y: 6² + y² = 100 → y = 8
Solve: 2(6)(1) + 2(8)(dy/dt) = 0 12 + 16(dy/dt) = 0 dy/dt = -12/16 = -0.75 m/s
Answer: Top slides down at 0.75 m/s.
Key insight: Use implicit differentiation and relate different rates of change.
What is an integral?
Geometric: Area under a curve Physical: Accumulation (total change)
Definite integral: ∫[a to b] f(x) dx = total area from x=a to x=b
Indefinite integral (antiderivative): ∫ f(x) dx = F(x) + C, where F'(x) = f(x)
Part 1: If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x)
(Derivative of integral is the original function)
Part 2: ∫[a to b] f(x) dx = F(b) - F(a), where F'(x) = f(x)
(To find area, find antiderivative and evaluate at endpoints)
Why this is profound: Derivatives and integrals are INVERSE operations.
∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1) ∫ 1/x dx = ln|x| + C ∫ e^x dx = e^x + C ∫ sin x dx = -cos x + C ∫ cos x dx = sin x + C
Don't forget +C! (Indefinite integrals have infinite antiderivatives)
1. Substitution (Reverse Chain Rule)
∫ 2x · e^(x²) dx
Let u = x², du = 2x dx ∫ e^u du = e^u + C = e^(x²) + C
2. Integration by Parts (Reverse Product Rule)
∫ u dv = uv - ∫ v du
Example: ∫ x·e^x dx Let u = x, dv = e^x dx Then du = dx, v = e^x ∫ x·e^x dx = x·e^x - ∫ e^x dx = x·e^x - e^x + C
LIATE rule (choosing u): Logarithmic > Inverse trig > Algebraic > Trig > Exponential
3. Partial Fractions
For rational functions (polynomial / polynomial): Break into simpler fractions, integrate separately.
Find area between y = x² and y = 2x from x=0 to x=2:
Setup: Top function: y = 2x Bottom function: y = x²
Area: ∫[0 to 2] (2x - x²) dx = [x² - x³/3] from 0 to 2 = (4 - 8/3) - (0 - 0) = 4/3
Disk method: Rotate around x-axis
V = π ∫[a to b] [f(x)]² dx
Shell method: Rotate around y-axis
V = 2π ∫[a to b] x·f(x) dx
Bad: "Chain rule is f'(g(x))·g'(x). I'll just apply it." Good: "Chain rule accounts for how BOTH functions change. Outer times inner derivative."
Test yourself: Can you explain the concept to a 10-year-old?
For derivatives: Graph the function, draw tangent lines For integrals: Shade the area you're finding For related rates: Draw and label the scenario
Visual understanding >> algebraic manipulation alone.
Process: 1. Attempt problem (15-20 min) 2. If stuck, look at HINT only 3. Try again (10 min) 4. Only then look at solution 5. Close solution, redo from scratch
Why? Struggling builds understanding. Looking at solutions too early = illusion of knowledge.
Bad: 6 hours before exam Good: 1 hour per day for a week
Spaced repetition >> cramming.
After each topic:
Problem: "I can do it in my head" Reality: Sign errors, dropped terms, wrong factors
Fix: Write every step, especially when learning.
Problem: You get a weird answer (negative area, velocity = 1000 m/s) and move on Fix: Sanity check. Does the answer make sense?
Problem: Seeing ∫ and using derivative rules Fix: Read the problem twice. Are you finding rate of change (derivative) or accumulation (integral)?
Problem: Losing points on every indefinite integral Fix: Write +C EVERY TIME (until it's automatic)
Problem: You know d/dx[sin x] = cos x but don't know d/dx[sin(2x)] Fix: Understand WHY (unit circle, rate of change). Then memorization is easier.
Free:
See your professor/TA if:
Fixed mindset: "I'm not a math person. I'll never get this."
Growth mindset: "I don't get this YET. I need to try a different approach."
Truth:
Calculus is beautiful when you understand it.
It's not about memorizing rules. It's about:
You've got this. 📈
