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Calculus 1 formula sheet

42 formulas · what each one means · a free lesson for each

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Limits

Direct substitution
limx → a f(x) = f(a)
Works when f is continuous at a.Lesson: Limits by algebra →
0/0: factor and cancel
limx → 2 x2 − 4x − 2 = limx → 2 (x + 2) = 4
Lesson: Limits by algebra →
Conjugate
multiply top and bottom by the conjugate
For a root in a 0/0 limit.Lesson: Limits by the conjugate →
Limits at infinity
degree top < bottom: 0 · equal: ratio of leading numbers
Lesson: Limits at infinity →
Continuity
limx → a f(x) = f(a)
f(a) exists, the limit exists, and they are equal.Lesson: Continuity →
L'Hôpital's rule
limx → a f(x)g(x) = limx → a f′(x)g′(x)
Only for 0/0 or ∞/∞.Lesson: L'Hôpital's rule →

Derivative rules

Definition
f′(x) = limh → 0 f(x + h) − f(x)h
The slope of the tangent line.Lesson: The derivative from the definition →
Power rule
ddx xn = nxn−1
Bring the power down, take 1 off the power.Lesson: The power rule →
Constant
ddx c = 0
Lesson: The power rule →
Constant multiple and sum
(cf)′ = cf′ (f + g)′ = f′ + g′
Lesson: The power rule →
Product rule
(fg)′ = f′g + fg′
Lesson: Product and quotient rules →
Quotient rule
(fg)′ = f′g − fg′g2
Lesson: Product and quotient rules →
Chain rule
ddx f(g(x)) = f′(g(x)) × g′(x)
Outside derivative × inside derivative.Lesson: The chain rule →

Derivatives of functions

sin and cos
ddx sin x = cos x ddx cos x = −sin x
Lesson: Derivatives of sin, cos, e and ln →
tan
ddx tan x = sec2 x
Lesson: Derivatives of sin, cos, e and ln →
eˣ and ln x
ddx ex = ex ddx ln x = 1x
Lesson: Derivatives of sin, cos, e and ln →
bˣ and log
ddx bx = bx ln b ddx logb x = 1x ln b
Lesson: Derivatives of bˣ / Derivatives of logarithms →
Inverse sine
ddx sin−1 x = 1√1 − x2
Lesson: Derivatives of inverse trig functions →
Inverse tangent
ddx tan−1 x = 11 + x2
Lesson: Derivatives of inverse trig functions →

Using derivatives

Implicit differentiation
ddx y2 = 2ydydx
Differentiate both sides; every y gets × dy/dx.Lesson: Implicit differentiation →
Tangent line
y − f(a) = f′(a)(x − a)
Lesson: Tangent lines →
Linear approximation
L(x) = f(a) + f′(a)(x − a)
Near a, the tangent line is close to f.Lesson: Linear approximation →
Differential
dy = f′(x) dx
Lesson: Differentials →
Critical points
f′(x) = 0 or undefined
Where a max or min can be.Lesson: Maximum, minimum and optimization →
Second derivative test
f″(c) > 0: min f″(c) < 0: max
At a point where f′(c) = 0.Lesson: The second derivative test →
Concavity
f″ > 0: up f″ < 0: down
Inflection point: where f″ changes sign.Lesson: Concavity and inflection points →
Mean value theorem
f′(c) = f(b) − f(a)b − a
Somewhere the slope equals the average slope.Lesson: The mean value theorem / Rolle's theorem →
Newton's method
xn+1 = xn − f(xn)f′(xn)
Lesson: Newton's method →
Logarithmic differentiation
ln y = ln f(x) → y′y = ddx ln f(x)
For powers like xˣ.Lesson: Logarithmic differentiation →

Basic integrals

Power rule
∫xn dx = xn+1n + 1 + C, n ≠ −1
Add 1 to the power, divide by the new power.Lesson: Antiderivatives →
1/x
∫1x dx = ln|x| + C
Lesson: Integrals with e and ln →
eˣ
∫ex dx = ex + C ∫ekx dx = 1kekx + C
Lesson: Integrals with e and ln →
sin and cos
∫sin x dx = −cos x + C ∫cos x dx = sin x + C
Lesson: Antiderivatives →
sec²
∫sec2 x dx = tan x + C
Lesson: Antiderivatives →
Constant multiple and sum
∫(af + bg) dx = a∫f dx + b∫g dx
Lesson: Antiderivatives →

Definite integrals

Fundamental theorem
∫ba f(x) dx = F(b) − F(a)
F is an antiderivative of f.Lesson: Definite integrals and area →
Fundamental theorem, part 1
ddx∫xa f(t) dt = f(x)
Lesson: The Fundamental Theorem, part 1 →
Riemann sum
Σ f(xi) Δx, Δx = b − an
Rectangles under the curve.Lesson: Riemann sums →
Area between curves
∫ba (top − bottom) dx
Lesson: Area between two curves →
Average value
1b − a∫ba f(x) dx
Lesson: Average value of a function →
Net change
∫ba F′(x) dx = F(b) − F(a)
Total change from the rate.Lesson: Net change →
Initial value problem
y = ∫f(x) dx, then find C from y(a)
Lesson: Initial value problems →

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