# Calculus Derivatives and Limits Math Sheet

An Engineers Quick Calculus Derivatives and Limits Reference

# Limits Math Help

## Definition of Limit

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.

## Right Hand Limit

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

## Left Hand Limit

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

## Limit at Infinity

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.

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

## Properties of Limits

Given the following conditions:

The following properties exist:

## Limit Evaluation Methods

### Continuous Functions

If f(x) is continuous at a then:

### Continuous Functions and Compositions

If f(x) is continuous at b:

# Derivatives Math Help

## Definition of a Derivative

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

## Mean Value Theorem

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

## Basic Properites

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

## Product Rule

The product rule applies when differentiable functions are multiplied.

## Quotient Rule

Quotient rule applies when differentiable functions are divided.

## Power Rule

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

## Chain Rule

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

## Chain Rule Examples

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