9. Derivatives

Calculus 1



Weiqi Zhou

Motivating Examples

Example 1: Tangent and Secant Lines

Example 2: Flying Arrow at Rest

More Examples

density、absorption rate etc.

Derivatives

if is defined in some neighbourhood of , then the following limit, if it exists, is called the derivative of at :

and will be denoted by or
in such case is also said to be differentiable at

It is also the slope of the tangent line of the curve at

One-sided Derivatives

left derivative:

right derivative：

existence of derivatives left and right derivative coincide

Example

  • compute the derivative of where is a constant



  • the derivative of a constant function is

Example

  • compute the derivative of where is a constant



Exercise

find the slope and the equation of the tangent line of at

Example

  • compute the derivative of



Example

  • compute the derivative of



Exercise

compute the derivative of

Exercise

decide whether is differentiable at and why

Continuous Functions Are Not Necessarily Differentiable

differentiability implies certain smoothness

Differentiable Functions Are Necessarily Continuous

if the derivative of exists at ， then must be continuous at

Proof



  • As ，



  • Consequently as , we have

Higher Derivatives

  • if is also diffentiable，then its derivative is called the second order derivative of , and will be denoted by
    ，or ，or

  • ，or is the notation for the -th derivative

  • may also be written as

Another Motivating Example

suppose that the side length of a square is increased
from to ,

then its area increases from to

Differentials

  • Given , write

  • If is differentiable at , then can be written into the form of

  • The infinitesimal change of as , is called the differential of at , and is denoted by

containn

differentiable: close to linear locally

hence it is good to adopt

for small

Summary

  • Derivatives

  • Computations of Derivatives

  • Differentiable and Continuity

  • Higher Derivatives and Differentials