Differential Calculus CheatSheet

Limits

The limit of a functionasapproaches a valueis denoted as:

whereis the value thatapproaches.

One-sided limits

One-sided limits are defined as:

• Left-hand limit:

• Right-hand limit:

Properties of limits

Sum

Difference

Product

4. Quotient (if):

Constants

Infinite limits

Limit asapproaches infinity

Limit at infinity of a rational function

If, whereandare polynomials, the limit can be evaluated as:

• If the degree ofis less than that of:

• If the degree ofis equal to that of:

• If the degree ofis greater than that of:

Indeterminate limits

Indeterminate limits are those that cannot be evaluated directly and require simplification. Some common indeterminate forms are:

L’Hôpital’s rule

L’Hôpital’s rule is used to resolve indeterminate limits of the formor:

if the limit on the right side exists.

Squeeze theorem

Iffor allin an interval containing(except possibly at), and

then:

Notable limits

Limit of

Limit of

Limit of

Limit of

Limits of trigonometric functions

Limit of

Limit of

Limit of series and sequences

Limit of an infinite series

Ifis a sequence, then:

Cauchy limit test

A sequenceconverges if, for every, there exists ansuch that:

Derivatives

Derivative of a functionat a point

If this limit exists,is said to be differentiable at.

Derivative of a function

Derivation rules

Sum rule

Ifandare differentiable functions, then:

Product rule

Ifandare differentiable functions, then:

Quotient rule

Ifandare differentiable functions and, then:

Chain rule

Ifandare differentiable functions, then:

Higher-order derivatives

Second derivative

Ifis differentiable, the second derivative ofis:

Derivative of order

The-th derivative of a functionis:

Important theorems

Rolle’s theorem

Ifis continuous on, differentiable on, and, then there exists asuch that:

Mean value theorem

Ifis continuous onand differentiable on, then there exists asuch that:

Inverse derivative theorem

Ifis differentiable and its inverseis also differentiable, then:

Applications of the derivative

Local maxima

A pointis a local maximum ofifand.

Local minima

A pointis a local minimum ofifand.

Inflection point

A pointis an inflection point if the concavity ofchanges at, that is, ifand.

Integration

Fundamental theorem of calculus

Ifis an antiderivative of, then:

Linearity property

Definition of improper integral

Integration rules

Sum rule

Product by constant rule

Integration methods

Substitution method

If, then:

Integration by parts method