# Differentiation.

*It's a warm summer evening in ancient Greece...*

::: author Apollonius of Perga A tangent to a curve C is a line l that intersects C, such that no other line could fall between l and C. :::

## Finding the tangent Using a limit.

- the line through P an Q is called a
**secant**. - the
**tangent**to the curve (at P) is the limit of the secant line for Q approaching P.

- the slope of the secant through P and Q is:

- The slope of the tangent to C at P is

## Differentiability and continuity

Definition

Let f be a function defined on an open interval I, and let

provided this limit exists.

If the limit exists we say: f is

**differentiable**atWhen a function is

**differentiable**it is also continuous.But a function can be

**continuous**but not**differentiable**.

### Possible notations for derivatives:

### Differentiation Rules 📜

- Sum Rule:
- Difference Rule:
- Constant multiple Rule:
- Product Rule:
- Quotient Rule:
- Chain Rule:

### Derivatives of elementary functions:

condition | ||
---|---|---|

c | 0 | |

## The tangent Line:

info

Let f be differentiable at a. The tangent line to the graph of f at P(a, f(a)) is given by the equation:

### Linearization

the function L is called the linearization of f at a. (it's the tangent line)

### Linearization as approximation:

- linearizations can be used as an approximation for a more complicated function.
- linearizations have less error the closer x is to the point at which you linearized (a in the tangent equation)

## Extreme Values:

Definition

Let f be a real function with domain D. Let

- f has an
**absolute maximum value**on D at c if:

- f has an
**absolute minimum value**on D at c if:

Extreme Value info

Conditions:

- function f is
**continuous**on interval [a, b] - interval [a, b] is
**closed**== (a and b are included in the interval) - interval [a, b] is
**bounded**== (a and b are real numbers)

Result:

- f attains both an absolute maximum value and an absolute minimum value on [a, b].

### Local Extrema:

Definition

Let f be a real function with domain D. Let

- f has an
**local maximum value**at c if there exists an open interval I containing c such that:

- f has an
**local minimum value**at c if there exists an open interval:

First Derivative info

If f has a local maximum or minimum value at an interior point c of it's domain, and f' is defined at c, then f'(c) = 0

If:

- c is not an interior point of I
- f is not differentiable at c. Or:
.

### Critical points

Definition

An interior point of the domain of f where f' is undefined or where f'(c) = 0 is called a critical point of f.

- A critical point is not necessarily a local extreme.

### Finding absolute extrema:

- Find all points (a, b) where f' does not exist.
- Find all
where f'(c) = 0. - Calculate the f(x) for x = [a, b, points that don't exist, points c where f'(c) = 0]
- The largest value is the absolute maximum value.
- The smallest value is the absolute minimum value.

## L'Hôpital's rule

### Indeterminate form

L'Hôpital's rule

Suppose that f(c) = g(c) = 0, that f and g are differentiable on an open interval I containing c and that

assuming that the limit on the right side of the equation exists.

- if the limit on the right hand side is of type
you can apply L'Hôpital again.

### Indeterminate form

- L'Hôpital's rule is also correct when

### Indeterminate form

For limits of the type:

### Indeterminate form and similar

If a limit of type

- first calculate
(hopital may be needed at this step) - then