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.

definition of a limit

Right Hand Limit

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 Limit

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 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.

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 Limits

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 +-Infinity

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 Methods

Continuous Functions

If f(x) is continuous at a then:

limit of a continuous function

Continuous Functions and Compositions

If f(x) is continuous at b:

limit of a the composition of continous functions

Factor and Cancel

limit evaluation method using factoring

L'Hospital's Rule

limit evaluation method using L'Hopital's rule

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:

definition of a derivative using limits

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.

derivatives mean value theorem

Basic Properites

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 Rule

The product rule applies when differentiable functions are multiplied.

derivative product rule - derivative of two functions multiplied

Quotient Rule

Quotient rule applies when differentiable functions are divided.

derivative quotient rule - derivative of the division of two functions

Power Rule

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 Rule

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

derivative of two functions applied to one another

Common Derivatives

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 Examples

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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