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Kaivan Karimi – Dir. of Global Strategy, Microcontroller Group, at Freescale

Calculus Derivatives and Limits Math Sheet

An Engineers Quick Calculus Derivatives and Limits Reference

$calculus derivatives limits sheet$

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'Hopital'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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