The 00 form is the basis of differentiation; the ∞×0 form is the basis of integration.
Limits and continuity
The limit of a function is the value that f(x) gets closer to as x approaches some number.
x→alimf(x)
"The limit of the function f as x approaches a." The symbol lim means we are taking the limit of something. The expression to the right is the expression we are taking the limit of. The expression x→a below lim means we are taking the limit of f as values of x approach a.
Approach
As values increase towards a, this is called approaching from the left (notated x→a−).
As values decrease towards a, this is called approaching from the right (notated x→a+).
When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.
Limit theorems
Arithmetic
The limit of a sum is the sum of the individual limits.
The limit of a difference is the difference of the individual limits.
The limit of a product is the product of the individual limits.
The limit of a quotient is the quotient of the individual limits, provided each individual limit exists and the limit of the denominator is nonzero.
Limits using direct substitution
Limits using algebraic manipulation
Limits by factoring
Limits by rationalizing
Two special limits
θ→0limθsin(θ)=1
θ→0limθcos(θ)=0
Derivatives
Definition of derivative
f′(x)=h→0limhf(x+h)−f(x)
Derivatives of trig functions
dxdsin(x)=cos(x)
dxdcos(x)=−sin(x)
dxdtan(x)=sec2(x)
dxdcot(x)=−csc2(x)
dxdsec(x)=sec(x)tan(x)
dxdcsc(x)=−csc(x)cot(x)
Derivatives of inverse trig functions
dxdsin−1(u)=1−u2du/dx
dxdcos−1(u)=1−u2−du/dx
dxdtan−1(u)=1+u2du/dx
dxdcot−1(u)=1+u2−du/dx
dxdsec−1(u)=∣u∣u2−1du/dx
dxdcsc−1(u)=∣u∣u2−1−du/dx
Derivatives of exponential functions
General rule for the derivative of exponential function: write it down x derivative of the exponent x natural log of the base
dxd8e2x=8e2x⋅2⋅ln(e)
Derivatives of logarithmic functions
General rule for the derivative of natural log functions: 1 over the object x derivative of the object
dxdln(u)=u1⋅dxdu
dxdlogax use the base-changing formula first
Integrals
Trigonometric power reduction identities
sin2(θ)=21−cos(2θ)
cos2(θ)=21+cos(2θ)
Integration of logarithmic trigonometric functions
∫tan(θ)dx=∫cos(θ)sin(θ)dx
∫sec(θ)dx⋅sec(θ)+tan(θ)sec(θ)+tan(θ)
∫csc(θ)dx⋅csc(θ)+cot(θ)csc(θ)+cot(θ)
Integration resulting in inverse trionometric functions