Calculus Formulas

Memorization of formulas for essentials of calculus

Slope of a line through (x₁, y₁) and (x₂, y₂)

m = (y₂ − y₁) / (x₂ − x₁)

Point-slope form of a line

y − y₁ = m(x − x₁)

Slope-intercept form of a line

y = mx + b

Slope of a horizontal line / vertical line

0 / undefined

Quadratic formula (roots of ax² + bx + c)

x = (−b ± √(b² − 4ac)) / 2a

When does a quadratic have no real roots?

When b² − 4ac < 0

a⁰ =

1

a^(1/n) =

ⁿ√a (the nth root of a)

a^(m/n) =

(ⁿ√a)^m

a^(−u) =

1 / a^u

a^u · a^v =

a^(u + v)

(a^u)^v =

a^(uv)

(ab)^u =

a^u · b^u

e^x = y is the same as…

x = ln y

ln 1 =

0

ln e =

1

ln(e^f(x)) =

f(x)

e^(ln f(x)) =

f(x)

Limit of a continuous function as x → a

lim f(x) = f(a) (just plug in a)

When does a two-sided limit exist?

When the left limit (x → a⁻) equals the right limit (x → a⁺)

Plugging in gives 0/0. What does it mean and what do you do?

Removable discontinuity: factor and cancel, use long division (with 0 placeholders), or use L'Hospital's rule

Limit definition of the derivative

f'(x) = lim (a → x) of [f(x) − f(a)] / (x − a)

Derivative of a constant c

0

Power rule: d/dx xⁿ

n·x^(n − 1)

Constant multiple rule: d/dx c·f(x)

c·f'(x)

Sum/difference rule: d/dx [f ± g]

f' ± g'

d/dx eˣ

eˣ

d/dx ln x

1/x

Tangent line to f(x) at x = a

y − f(a) = f'(a)(x − a)

Product rule: d/dx [f·g]

f·g' + g·f'

Quotient rule: d/dx [f/g]

(g·f' − f·g') / g² ("low d-high minus high d-low, over low squared")

Chain rule: d/dx f(g(x))

f'(g(x)) · g'(x)

d/dx e^g(x)

e^g(x) · g'(x)

d/dx ln g(x)

g'(x) / g(x)

d/dx [g(x)]ⁿ

n·[g(x)]^(n − 1) · g'(x)

f'(x) > 0 means…

f is increasing

f'(x) < 0 means…

f is decreasing

Critical point

A point where f'(x) = 0

f' changes from + to −

Local maximum

f' changes from − to +

Local minimum

Second derivative f''(x)

The derivative of f'(x)

f''(x) > 0 means…

Convex (opens up like a cup / smile)

f''(x) < 0 means…

Concave (opens down like a cap / frown)

Second derivative test: f'(a) = 0 and f''(a) > 0

Local minimum at x = a

Second derivative test: f'(a) = 0 and f''(a) < 0

Local maximum at x = a

Second derivative test: f'(a) = 0 and f''(a) = 0

No conclusion

Closed interval method on [a, b]

Find critical points inside [a, b]; evaluate f (not f') at them and at a and b; largest = global max, smallest = global min

Total revenue R(x)

R(x) = x · P(x)

Profit Π(x)

Π(x) = R(x) − C(x)

Marginal cost / marginal revenue

C'(x) / R'(x)

Profit-maximizing condition

Π'(x) = 0, which means R'(x) = C'(x)

R(x) = C(x) means…

Break-even point (zero profit), not maximum profit

L'Hospital's rule (when plugging in gives 0/0)

lim f(x)/g(x) = lim f'(x)/g'(x) (differentiate top and bottom separately)

Newton's method formula

x(n+1) = x(n) − f(x(n)) / f'(x(n))

When does Newton's method fail?

When f'(xₙ) = 0 (horizontal tangent), or the estimates drift away or oscillate