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.
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.
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.
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.
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.
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.
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.
📐 Calculus
"UV Voodoo" — ∫u dv = uv − ∫v du
Integration by Parts
The weirdest-sounding mnemonic that actually works
"UV Voodoo" — u·v minus the integral of v·du. Use when substitution won't work. Use LIATE to choose u.
📐 Calculus
Concave Up = Cup ☕. Concave Down = Frown ☹
Concavity
Never mix up concave up and concave down again
Concave up looks like a cup ☕. Concave down looks like a frown ☹. f''(x) > 0 = concave up. f''(x) < 0 = concave down.
📐 Calculus
L'Hôpital: 0/0 or ∞/∞? Differentiate top and bottom separately
L'Hôpital's Rule
The rule that rescues you from indeterminate forms
When limit gives 0/0 or ∞/∞, differentiate numerator AND denominator separately (NOT quotient rule!), then evaluate again.
📐 Calculus
Critical Points: f'(x) = 0 → test sign change
Finding Maxima & Minima
Where maxima, minima and saddle points hide
Set f'(x) = 0. f' + to − = local MAX. f' − to + = local MIN. No sign change = saddle point.