# Derivatives

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## Derivative of a function *f* at a number *a*

edit
### Notation

editWe denote the derivative of a function at a number as .

### Definition

editThe derivative of a function at a number a is given by the following limit (if it exists):

An analogous equation can be defined by letting . Then , which shows that when approaches , approaches :

### Interpretations

edit#### As the slope of a tangent line

editGiven a function , the derivative can be understood as the slope of the tangent line to at :

#### As a rate of change

editThe derivative of a function at a number can be understood as the instantaneous rate of change of when .

## At a tangent to one point of a curve

edit### Vocabulary

editThe point A(a; f(a)) is the point in contact of the tangent and Cf.

### Definition

editIf f is differentiable in a, then the curve C admits at a point A which has for coordinates (a ; f(a)), a tangent : it is the straight line passing by A and of direction coefficient f'(a). An equation of that tangent is written: y = f'(a)*(x-a)+f(a)

## Degrees

edit### First Degree Derivative [First Order Derivative; f'(x)]

editThe first degree derivative of a function, commonly showing the slope of the tangent line at one point of the function, shows its instantaneous rate of change. Intuitively, the first degree derivative reveals the direction of the function; a positive first degree derivative shows the increasing of a function, and it shows decreasing when negative.

## Minimum/Maximum

edit##### Local

editLocal minimums/maximums are found from solving for **f'(x)=0**, for f(x) is differentiable for all x on desired interval. When the first order derivative is zero, it suggests the stop in increasing or decreasing. Whether the point x at f'(x)=0 is a maximum or minimum requires the derivative to the left and right of x. If the derivative is positive on the left and negative on the right; if the graph shifts from increasing to decreasing at point x, f(x) at x would be a **local maximum**. Conversely, If the derivative is negative on the left and positive on the right; if the graph shifts from decreasing to increasing at point x, f(x) at x would be a **local maximum**.

##### Global

editGlobal minimums/maximums are found at the smallest/largest y-values of each graph.

A **saddle point** refers to the point where f'(x)=0 but the graph maintains its movement direction.

#### Application

editIn a function that measures an object's disposition, the first order derivative shows its instantaneous change in position, i.e. **velocity**.

### Second Degree Derivative [Second Order Derivative; f''(x)]

editThe second order derivative takes

#### Inflection Point

editAn inflection point refers to the point where the function changes its concavity (from sloping up to down, vice versa). The inflection point is found from solving for **f''(x)=0**, for f(x) is differentiable for all x on desired interval.

#### Application

editIn a function that measures an object's velocity, the first order derivative shows its instantaneous change in velocity, i.e. **acceleration**.

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## Differentiation Rules

edit### Power Rule

editOne of the most commonly used derivative rules for functions in the form of ( raised to the th power), .

#### Polynomial Differentiations

editTaking the derivative of a polynomial requires the differentiation procedure to be applied to each term including the variable .

Example:

#### Negative Power Differentiations

editTaking the derivative of a function in the form can be achieved through rewriting the function with a negative power, .

Example:

#### Fractional Exponents and Radicals

editDifferentiating a function with a radical such as a square root, , can be done through rewriting the function in the form with a fraction as the exponent, .

Example:

### Quotient Rule

editThis rule is used to differentiate functions written in the form of .

Example:

### Trigonometric Functions

editRules for differentiating trigonometric functions:

### Logarithmic Functions

editRules for differentiating logarithmic functions:

## Sample Problems

editDifferentiate the following: