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∫ Calculus · Derivatives
Low D-High minus High D-Low, over Low-Low
Quotient Rule — (f/g)' = (g·f' − f·g') / g²
Never mix up the quotient rule numerator again
"Low D-High minus High D-Low, square the bottom and away we go." Low = denominator (g), High = numerator (f), D = derivative. Result: (g·f' − f·g') / g².
Differentiate composite functions without missing a step
Derivative of the outside (leave inside alone) times derivative of the inside. Example: d/dx[sin(x²)] = cos(x²) · 2x. Always multiply by the inner derivative — the most missed step!
Step 1
Identify the outer function f and inner function g(x)
Step 2
Differentiate the outer function, leaving the inner unchanged: f'(g(x))
Step 3
Multiply by the derivative of the inner function: × g'(x)
L'Hôpital's Rule — L=L'Hôpital · I=If · P=Problem is 0/0 or ∞/∞ · S=Separately differentiate top and bottom
Escape 0/0 and ∞/∞ indeterminate forms instantly
If lim f(x)/g(x) = 0/0 or ∞/∞, differentiate numerator and denominator SEPARATELY (not the quotient rule!) and re-evaluate. Repeat if still indeterminate.
L
L'Hôpital's Rule — named after Guillaume de l'Hôpital (1661–1704)
I
If — only apply the rule when an indeterminate form exists
P
Problem — the limit evaluates to 0/0 or ∞/∞
S
Separately — differentiate top and bottom independently, not as a quotient
Integration by Parts u-choice — L=Logarithmic · I=Inverse trig · A=Algebraic · T=Trigonometric · E=Exponential
Always know which term to call "u" in integration by parts
∫u dv = uv − ∫v du. Choose u from whichever LIATE category appears first. Log functions first, exponentials last — because logs simplify when differentiated, exponentials don't change.
L
Logarithmic — ln(x), log(x) — choose as u first; derivative simplifies nicely
I
Inverse trig — arcsin, arctan, arccos — choose second
A
Algebraic — x, x², polynomials — choose third
T
Trigonometric — sin, cos, tan — choose fourth
E
Exponential — eˣ, 2ˣ — choose last; use as dv since it integrates to itself
Product Rule: First × D(Second) + Second × D(First)
Product Rule — d/dx[f·g] = f·g' + g·f'
Differentiate products of two functions correctly every time
"First times derivative of second, plus second times derivative of first." d/dx[x²·sin(x)] = x²·cos(x) + sin(x)·2x. Never just multiply the two derivatives — that is always wrong!
Squeeze Theorem: g≤f≤h and lim g=lim h=L → lim f=L
Squeeze Theorem — trap the function between two known limits
Prove limits of tricky functions by trapping them
Classic: lim(x→0) x²sin(1/x)=0 because −x²≤x²sin(1/x)≤x² and both bounds go to 0. The function is squeezed to 0 even though sin(1/x) oscillates wildly.
U-Substitution — the Chain Rule in reverse for integration
The most powerful basic integration technique
See a composite function multiplied by the inner function's derivative? Set u=inner function, du=inner derivative × dx, replace everything, integrate in u, substitute back. Example: ∫2x·cos(x²)dx → u=x², du=2x dx → ∫cos(u)du=sin(x²)+C.
Three Things Students Forget — C=Critical points · A=Always check endpoints · N=Never skip +C
Three things calculus students always lose points on
Critical points: set f'(x)=0 or undefined. Absolute extrema on [a,b]: always test endpoints too — the max or min might be there! Indefinite integrals: always add +C or lose points every time.
C
Critical points — set f'(x)=0 or find where f' is undefined; these are candidates for extrema
A
Always check endpoints — on a closed interval [a,b], absolute max/min might be at a or b, not just critical points
N
Never forget +C — every indefinite integral needs a constant of integration or the answer is incomplete
Formal Limit Definition — for every ε>0 there exists δ>0 such that whenever |x−a|<δ then |f(x)−L|<ε
The formal definition of a limit — what it really means
For every tiny output tolerance ε, you can find an input restriction δ so that whenever x is within δ of a, f(x) is within ε of L. Think: guaranteed output closeness from input closeness.
0/0 or infinity/infinity? Differentiate TOP and BOTTOM separately. Check the form FIRST.
Resolving indeterminate forms by differentiating numerator and denominator independently
Only apply when you have 0/0 or infinity/infinity — verify the form before applying
If the limit gives 0/0 or infinity/infinity, differentiate numerator and denominator separately and try again. Repeat if still indeterminate. Other forms: rewrite 0 times infinity as a fraction first. For 1 to the infinity, 0 to the 0, or infinity to the 0: take ln first. Never apply when the form is not indeterminate.
Check form first
Must be 0/0 or infinity/infinity before applying
Differentiate separately
Top derivative over bottom derivative — not the quotient rule
Repeat if needed
Still indeterminate after one application? Apply again.
Differentiate both sides with respect to x. Every y-term gets multiplied by dy/dx from the chain rule.
Finding dy/dx when y cannot be isolated — chain rule applied to y as a function of x
Treat y as a function of x — every y-term gets the chain rule factor dy/dx
Steps: Differentiate both sides with respect to x. Every y-term gets multiplied by dy/dx. Collect all dy/dx terms on one side. Factor out dy/dx. Divide to solve. Example: x squared plus y squared equals 25 gives 2x plus 2y times dy/dx equals 0, so dy/dx equals negative x over y. Use for circles, ellipses, and equations where y cannot be isolated.
Differentiate all
Every term on both sides gets differentiated with respect to x
y gets dy/dx
Chain rule: d/dx of y squared equals 2y times dy/dx
WIRED — Write objective, Identify constraint, Reduce to one variable, Extremize, Decide max or min.
Five-step approach to every calculus optimization problem
Always verify it is a maximum or minimum using the second derivative or endpoint comparison
Write the objective function (what to maximize or minimize). Identify the constraint equation. Reduce to one variable using the constraint. Find critical points by setting the derivative to zero. Decide max or min using the second derivative test or endpoint comparison. Always state the actual max or min value, not just where it occurs.
WIRED
Write, Identify, Reduce, Extremize, Decide
One variable
Use constraint to substitute and reduce before differentiating
Verify
Second derivative test or endpoint check confirms max vs min
Draw it. Label it. Write geometric equation. Differentiate with respect to TIME. Plug in values LAST.
Finding how fast one quantity changes given the rate of change of a related quantity
Never substitute changing values BEFORE differentiating — only plug in after taking the derivative
Draw and label a diagram. Identify the given rates and the rate you want. Write an equation relating the variables. Differentiate implicitly with respect to time — every variable becomes a rate like dx/dt or dy/dt. Then plug in the values at the specific moment. Common formulas: volume of sphere equals four thirds pi r cubed, ladder uses x squared plus y squared equals L squared.
Draw first
Diagram with labeled variables prevents setup errors
Differentiate w.r.t. t
Every variable becomes a rate: dx/dt, dy/dt, dr/dt
Check Divergence Test first. Geometric or p-series use formulas. Factorials or exponentials use Ratio Test.
Which convergence test to apply for each type of series
If the limit of terms is not zero, the series diverges immediately — always check this first
Divergence Test first: if limit of terms is not zero, series diverges. Geometric: converges if absolute value of r is less than 1, sum equals a over (1 minus r). p-series 1 over n to the p: converges if p is greater than 1. Ratio Test: compute limit of absolute value of (a subscript n+1 over a subscript n) — less than 1 converges, greater than 1 diverges. Best for factorials and exponentials.
Divergence Test
Always check first — if terms do not approach zero, done
Ratio Test
Best for n factorial and a to the n — compute limit of ratio
p-series
1 over n to the p converges if p is greater than 1
Disk: pi times integral of f(x) squared. Washer: pi times integral of outer squared minus inner squared. Shell: 2pi times integral of x times f(x).
Three methods for volume when a region is rotated about an axis
Disk and Washer integrate perpendicular to the axis. Shell integrates parallel to the axis.
Disk method for solid with no hole: V equals pi times integral of f(x) squared dx. Washer method for region between two curves: V equals pi times integral of (outer squared minus inner squared) dx. Shell method: V equals 2pi times integral of x times f(x) dx — useful when rotating about the y-axis. Both methods give the same answer so choose the simpler integral.
Disk
Pi times integral of f(x) squared dx — solid, no hole
Washer
Pi times integral of (outer squared minus inner squared) dx
Shell
2pi times integral of x times f(x) dx — cylindrical shells
Q: What does LIATE stand for and how does it determine which term is u in integration by parts?
A: LIATE: Logarithmic · Inverse trig · Algebraic · Trigonometric · Exponential. Choose u from whichever category appears first in LIATE. Logs are chosen first because they simplify when differentiated. Exponentials are chosen last (used as dv) because eˣ integrates to itself. Example: ∫x·eˣ dx → x is Algebraic (A), eˣ is Exponential (E) → u=x, dv=eˣdx → du=dx, v=eˣ → answer: xeˣ−eˣ+C.
Q: State the Chain Rule and give two examples.
A: d/dx[f(g(x))] = f'(g(x)) · g'(x). Always multiply by the derivative of the inner function. Example 1: d/dx[sin(x²)] — outer is sin (derivative: cos), inner is x² (derivative: 2x) → cos(x²)·2x. Example 2: d/dx[(3x+1)⁵] — outer is ( )⁵ (derivative: 5( )⁴), inner is 3x+1 (derivative: 3) → 15(3x+1)⁴. Most common mistake: forgetting to multiply by the inner derivative.
Q: State both parts of the Fundamental Theorem of Calculus.
A: Part 1: If F(x)=∫ₐˣ f(t)dt, then F'(x)=f(x) — differentiating an integral with variable upper limit returns the original function. Part 2: ∫ₐᵇ f(x)dx = F(b)−F(a) — evaluate a definite integral by finding any antiderivative and subtracting its values at the bounds. Together they show differentiation and integration are inverse operations — the central insight of calculus.
Q: When does L'Hôpital's Rule apply and how do you use it?
A: LIPS — L'Hôpital If Problem is 0/0 or ∞/∞. Step 1: evaluate the limit and confirm it gives 0/0 or ∞/∞. Step 2: differentiate the numerator and denominator SEPARATELY — do NOT use the quotient rule. Step 3: evaluate the new limit. Repeat if still indeterminate. Example: lim(x→0) sin(x)/x = 0/0 → differentiate → cos(x)/1 → cos(0)/1 = 1. Warning: does NOT apply to 0/1, 1/0, or other non-indeterminate forms.
Q: What is u-substitution and how do you recognize when to use it?
A: U-substitution reverses the chain rule for integration. Recognize it when you see a composite function multiplied by the derivative of the inner function. Steps: (1) Let u=inner function. (2) du=inner derivative·dx. (3) Rewrite entire integral in terms of u. (4) Integrate with respect to u. (5) Substitute back. Example: ∫2x·cos(x²)dx → u=x², du=2x dx → ∫cos(u)du = sin(u)+C = sin(x²)+C. Always add +C for indefinite integrals.