derivative by definition.pdf

7/27/2019 Derivative by Definition.pdf

1/4

CalculusPre Calculus

0

derivative= slopeof a curve at aspecific point.

How do you find slope if you onlyknow 1 point ???

In the prior lesson on "what is a derivative", we spent a lot of timetrying to understand the difference between a secant and a tangent

and, at the end, we learned that a derivative is the slope of a line ata specific point. However, there is a major problem here:

A Sp e c i f i c Ex am p l e o f o u r d i l e mm a Imagine that we have some curve, like the red one below, and we

know that the point (1,2) is on that curve. The slope of the purpleline is the derivative so all that we need to know is the slope of thepurple line.

But how do we find the slope of a line when we only know 1 point ?

Derivative by definition

Homecalculus >derivative >

12

Like

Send

slope = = y x

2 1

Determinant

(Coming Soon)

What isCalculus?

DerivativesIntegrals (coming soon)

Differential EquationsSequences & SeriesCalculus CalculatorGraping Calc

Geometric Series

Page 1 of 4Derivative by Definition-Formal definition explained with examples, pictures, and an ...

2013-05-03http://www.mathscoop.com/calculus/derivatives/derivative-by-definition.php


2/4

means the

'slope of thetangent line at x="a" '

0

0

H o w d o w e d e f i n e t h e t a n g e n t

m a t h e m a t i ca l l y ? There are two common definitions of tangent. Both of these

definitions are equivalent definitions of the derivative at a point. Theyare presented side by side below.

One of the notations used to represent the derivative is

Reset Both Pause Both

Version 1

Version 2

This version represents the

derivative with respect to thepoint

Wh a t d o e s t h is m e a n ?

The formula above meansFind the slope between the

points (x,f(x)) and (a,f(a)) suchthat the point (x,f(x)) getsinfinitesimally close to (a,f(a))

In this equation, we are saying

find the slope of the tangent linebetween two points when youadd almost zero to go from the

first value to the next.

Wh a t d o e s t h i s m e a n ?

This formula represents the

same idea as version 1. We aresaying that the points arepractically identical because we

only moved an extremely smallamount over. Point one

and point two

so the

Example Problem 1

Find the slope of the tangent line of using both formulasat x = 5

Version 1 Version 2

Example Problem 2Find the slope of the tangent line of using both formulas

at x = 2

Version 1 Version 2

Example Problem 3Find the slope of the tangent line of using both formulas at x

= 3

Version 1 Version 2

12

Like

Send

f

(a,f(a)),(i.e. (a))f

(a)= f lim x a

f(x) f(a)

a

x0

)= = f 0 lim h 0

f( + h) f )x0 x0

h

( f( ))x0 x0( + h,f( + h))0 0 x = ( + h) = h0 0

Like 12 Send

y = + 3xx2

y = 4x + 1

y = 1x 1




3/4

Practice Problems

Take Quiz

Next Lesson:

Derivative: General Definition

General Formula of the DerivativeThe following definition is an extension ofversion 2 for any generalpoint 'x' (instead of the specific value ).

Th e d e r i v a t i v e w i t h r e s p e c t

t o x

This formula tells us thederivative of the function f(x), we can then use the answer to find tehslope at any point without have to keep repeating the formula .

0

GraphingCalculator

Calculus HomeworkAnswers

Derivative Calculus Math Formula

x0

(x)= = f lim h 0f(x+ h) f(x)

h

12

Like

Send




4/4

0

Student ResourcesCopyright & Terms of Use

Site MapCalculus Homework Answers

Math Games OnlineOnline Calculator

2010 by Math Scoop.

Like 12 Send


2013 05 03http://www mathscoop com/calculus/derivatives/derivative by definition php

derivative by definition.pdf

Documents