Calculus Derivatives and Limits Math Sheet

An Engineers Quick Calculus Derivatives and Limits Reference

Limits Math Help

Definition of LimitReturn to Top

The limit is a method of evaluating an expression as an argument approaches a value. This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity. The following expression states that as x approaches the value c the function approaches the value L.

definition of a limit

Right Hand LimitReturn to Top

The following expression states that as x approaches the value c and x > c the function approaches the value L.

right hand limit definition

Left Hand LimitReturn to Top

The following expression states that as x approaches the value c and x < c the function approaches the value L.

left hand limit definition

Limit at InfinityReturn to Top

The following expression states that as x approaches infinity, the value c is a very large and positive number, the function approaches the value L.

limit at infinity

Also the limit as x approaches negative infinity, the value of c is a very large and negative number, is expressed below.

limit at negative infinity

Properties of LimitsReturn to Top

Given the following conditions:

conditions for limit properties

The following properties exist:

limit property with constant

limit property with the sum of two functions

limit property with the multiplcation of two functions

limit propety with the division of two functions

limit property with a function raised to a power

Limit Evaluation at +-InfinityReturn to Top

limit of e raised to the x at infinity

limit of the natural log at infinity

limit of a constant over x raised to a constant

limit of a constant over x raised to a constant when x raised r is real

limit of x raised to a constant for even r

limit of x raised to a constant for odd r

Limit Evaluation MethodsReturn to Top

Continuous FunctionsReturn to Top

If f(x) is continuous at a then:

limit of a continuous function

Continuous Functions and CompositionsReturn to Top

If f(x) is continuous at b:

limit of a the composition of continous functions

Factor and CancelReturn to Top

limit evaluation method using factoring

L'Hospital's RuleReturn to Top

limit evaluation method using L'Hopital's rule

Derivatives Math Help

Definition of a DerivativeReturn to Top

The derivative is way to define how an expressions output changes as the inputs change. Using limits the derivative is defined as:

definition of a derivative using limits

Mean Value TheoremReturn to Top

This is a method to approximate the derivative. The function must be differentiable over the interval (a,b) and a < c < b.

derivatives mean value theorem

Basic ProperitesReturn to Top

If there exists a derivative for f(x) and g(x), and c and n are real numbers the following are true:

derivative of a function with a constant

derivative of the sum of two functions

derivative of a constant

Product RuleReturn to Top

The product rule applies when differentiable functions are multiplied.

derivative product rule - derivative of two functions multiplied

Quotient RuleReturn to Top

Quotient rule applies when differentiable functions are divided.

derivative quotient rule - derivative of the division of two functions

Power RuleReturn to Top

The power rule applies when a differentiable function is raised to a power.

derivative power rule- derivative of a function raised to the power

Chain RuleReturn to Top

The chain rule applies when a differentiable function is applied to another differentiable function.

derivative of two functions applied to one another

Common DerivativesReturn to Top

derivative of a variable

derivative of the sin function

derivative of the cosine function

derivative of the tangent function

derivative of the secant function

derivative of the cosecant function

derivative of the cotangent function

derivative of the inverse sine function

derivative of the inverse cosine function

derivative of the inverse tangent function

derivative of a constant raised to variable

derivative of e raised to the power of x

derivative of the natural log function

derivative of the natural log absolute value function

derivative of the log function

Chain Rule ExamplesReturn to Top

These are some examples of common derivatives that require the chain rule.

chain rule example with function raised to power

chain rule example with e raised to a function

chain rule example of the natural log of function

chain rule example of the sin of a function

chain rule example of the cosine of a function

chain rule example of the tangent of a function

chain rule example of the secant of a function

chain rule example of the inverse tangent of a function

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