The form is the basis of differentiation; the form is the basis of integration.
Limits and continuity
The limit of a function
The limit of a function is a tool for investigating the behavior of as approaches some number .
Read as "the limit of the function as approaches ." The symbol means we are taking the limit of something. The expression to the right is the expression we are taking the limit of. The expression below means we are taking the limit of as values of approach .
As values increase towards , this is called approaching from the left (notated ).
As values decrease towards , this is called approaching from the right (notated ).
When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.
Formal definition of a limit
means that to establish a specific limit, a number is chosen first to establish a desired degree of proximity to , then a number is found that determines how close must be to to ensure that is within units of .
Algebraic computation of limits
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.
Two special limits
Exponential and logarithmic functions
Definition of derivative
Procedural rules for finding derivatives
See differentiation formulas.
Derivatives of trig functions
Derivatives of inverse trig functions
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
Derivatives of logarithmic functions
General rule for the derivative of natural log functions: 1 over the object x derivative of the object
- use the base-changing formula first
Applications of derivatives
A function is an antiderivative of on an interval if
for all in .
If is an antiderivative of , then so is for any constant given that the derivative of any constant is .
Indefinite integral notation
The antiderivative (integral) of is .
- is the integrand
- is the variable of integration
See integration formulas
To obtain the power rule, for is any number and
Integration by substitution
Integration by substitution or u-substitution is the inverse of the chain rule for diferrentiation.
Let , , and be differentiable functions of such that
Where is an antiderivative of .
Trigonometric power reduction identities
Integration of logarithmic trigonometric functions
Integration resulting in inverse trionometric functions