The precise formal definition of the derivative avoids the term "instantaneous
rate" by working directly in terms of the limit
of average rates (which are themselves just ratios of differences).

## DefinitionFor any function, f , its derivative is the function, f ', whose value at any point x is given by |

In the above definition the symbol h refers to a quantity that is being
taken to a particular limiting value (in this case 0), and since h is not
a value on which the limit depends, it is often called a "dummy variable"
and could be replaced by any other symbol without affecting the result.
A popular choice is to use instead the expression 'delta x' to emphasize
that it referrs to a **d**ifference of x values.

Another equivalent version of the definition is to work in terms of the displaced point x+h (let's call it z) rather than the displacement h, and to consider the limit as z->x of the ratios (f(z)-f(x))/(z-x)

In the above limit, the values of h can be either positive or negative, and for f to be "differentiable" at x it is necessary for this two sided limit to exist. But we could also consider the "one-sided derivatives" that one might define by looking at just positive or just negative values of h. This concept is illustrated at derivOneSide@ies

The name f ' that we have used for the derivative of f is
similar to that used by Newton (except that he used a dot directly over
the f rather than a 'prime' or apostrophe).

Another notation that is often used is due to Leibnitz, the other main
inventor of Calculus.

Instead of f', Leibnitz wrote df/dx (which is not intended to make
sense of df and dx separately, but just to remind us of the definition
as a limit of ratios). One problem with Leibnitz' notation is that it depends
on having a name for the input variable (x), and that is not really a property
of the function f itself, as any real value in its domain can be used as
input. This can sometimes be a source of confusion, especially when dealing
with composite functions and the chain rule, so watch out!

Some modern texts use the notation Df rather than f ' . This emphasizes
the idea of differentiation as an operation on the function f, and allows
us to refer to that operation itself with the symbol D.

When we want to emphasize differentiation as an operation, in terms
of Leibnitz' notation we often write df/dx as (d/dx)f, so the differentiation
operator D above can be written as d/dx.

If you have come across any good web-based discussions of the definition
of the derivative,

please do let
us know and we will add them here.

Review Contents of Math&Stats Dep't Website | ....... | Give Feedback | ....... | Return to Langara College Homepage |

- Introduction to Derivative Concept (@ubc)
- Intro to Derivative from Hofstra project
- Derivatives Intro @PWS
- Surfing(Derivatives)@ies
- VisualCalculus - Tangent Lines
- Animation:all the tangent lines to a curve
- Deriv from TanSlope(MathCad+QT@odu)
- (graphics.html#tangent)byDougArnold@pennState
- (graphics-j.html#tangent)byDougArnold@pennState
- MathsOnlineGallery (@uVienn.austria) derivative(asTanSlope)
- Why Slopes -- A Calculus Preview for Algebra Students
- Derivatives-- Introduction -- Curved Mirrors
- Stressed Out - Slope as Rate of Change
- derivDef@ies
- Sec&TanLines@ies
- DerivativeDefinition Java Applet
- JavaDiff(by kThompson@OregonState)
- Animation:Secant approaching the Tangent
- Tangent Line Applet byDanSloughter@furman
- SecantGrapher byJohnOrr@uNebraskaLincoln
- Visual Calculus - Tangent as Limit of Secants
- (graphics.html#secants)byDougArnold@pennState
- (graphics-j.html#secants)byDougArnold@pennState
- Our Own Version of the Secant -> Tangent thing
- Affine Approx Applet byDanSloughter@furman
- derivOneSide@ies
- Rolling Ball Applet (@uPenn)
- Velocity@upenn(withQTmovies)
- Relative Motion@upenn
- (graphics.html#bounce)byDougArnold@pennState
- (graphics-j.html#bounce)byDougArnold@pennState
- Derivatives: Rate of change/Learn
- Avg&InstRates - from PWS
- Calculus Online@ubc: Lab 2(avg&inst rates)
- Lab 2 Grades and Solutions
- Continuity and Differentiability(@ubc)
- MathsOnlineGallery (@uVienn.austria) NowhereDiffble
- (graphics-j.html#jagged)byDougArnold@pennState
- (graphics.html#jagged)byDougArnold@pennState
- Calc 1 pathologies(byTVogel@TexA&M)
- What does the derivative tell us about a function?(@ubc)
- Pickthe correct derivative! by David Yuen
- CurvatureCircle@ies
- LnGraphCircle@ies
- ProdRule@ies
- derivCubics@ies
- Introduction to Derivatives Computation(@ubc)
- Differentiating Linear Functions
- Differentiating Sums(@ubc)
- Differentiating Products(@ubc)
- Differentiating Quotients(@ubc)
- PrecalcExp&Logs@langara
- Karl's Calculus Tutor - 6.1 Be Fruitful and Multiply
- S.O.S. Math - Calculus(starts with exp&log)
- e(1)@ies
- TheNumber e (def by exp'(0)=1)
- Derivatives of Exponentials(@ubc)
- PrecalcTrigFunctions@langara
- DerivSin@ies
- TheDerivetaive of the Sine
- derivSinAnimation@odu
- Derivatives of Trigonometric functions(@ubc)
- Inverse Trigonometric Functions(@ubc)
- Relative Motion@upenn
- Composite Functions@ies
- ChainRule@ies
- The Chain Rule(@ubc)
- Implicit Differentiation(@ubc)
- Applications of The Chain Rule(@ubc)

- MVT@ies