Differential Calculus CheatSheet

Limits

The limit of a function as approaches a value is denoted as:

where is the value that approaches.

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 as approaches infinity

Limit at infinity of a rational function

If , where and are polynomials, the limit can be evaluated as:

• If the degree of is less than that of :

• If the degree of is equal to that of :

• If the degree of is 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 form or :

if the limit on the right side exists.

Squeeze theorem

If for all in 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

If is a sequence, then:

Cauchy limit test

A sequence converges if, for every , there exists an such that:

Derivatives

Derivative of a function at a point

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

Derivative of a function

Derivation rules

Sum rule

If and are differentiable functions, then:

Product rule

If and are differentiable functions, then:

Quotient rule

If and are differentiable functions and , then:

Chain rule

If and are differentiable functions, then:

Higher-order derivatives

Second derivative

If is differentiable, the second derivative of is:

Derivative of order

The -th derivative of a function is:

Important theorems

Rolle’s theorem

If is continuous on , differentiable on , and , then there exists a such that:

Mean value theorem

If is continuous on and differentiable on , then there exists a such that:

Inverse derivative theorem

If is differentiable and its inverse is also differentiable, then:

Applications of the derivative

Local maxima

A point is a local maximum of if and .

Local minima

A point is a local minimum of if and .

Inflection point

A point is an inflection point if the concavity of changes at , that is, if and .

Integration

Fundamental theorem of calculus

If is 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