Memory tricks for calculus

"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.

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!

d/dx[x⁵]=5x⁴. d/dx[x³]=3x². d/dx[x]=1. d/dx[constant]=0. Bring the exponent down as a coefficient, then reduce the exponent by one.

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.

∫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.

Part 1: d/dx[∫ₐˣ f(t)dt] = f(x). Part 2: ∫ₐᵇ f(x)dx = F(b)−F(a), where F is any antiderivative of f. Integration and differentiation undo each other.

"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!

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.

d/dx[sin x]=cos x → d/dx[cos x]=−sin x → d/dx[−sin x]=−cos x → cycle repeats every 4. Also: tan'=sec², cot'=−csc², sec'=sec·tan, csc'=−csc·cot.

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.

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.

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.

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.

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.

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.

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.

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.

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.