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Differentiation

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Presentation Transcript
slide1

A formal method of finding a gradient function.

y

average gradient…

f(x+h)-f(x)

h

x

x+h

x

slide2

y

f(x+h)-f(x)

h

x

x+h

x

slide3

y

f(x+h)-f(x)

h

x

x

x+h

slide4

y

f(x+h)-f(x)

h

x

x

x+h

slide5

y

f(x+h)-f(x)

h

x

x

x+h

slide6

y

f(x+h)-f(x)

h

x

x

x+h

slide7

y

f(x+h)-f(x)

h

x

x

x+h

slide8

y

f(x+h)-f(x)

h

x

x

x+h

slide9

y

x

x

x+h

slide10

y

x

x

x+h

slide11

y

instantaneous gradient at x

x

x:x+h

slide12

As h→0, the average gradient approaches the actual gradient at x

Gradient at x

As

This leads to the formal definition of the derivative

slide13

Definition: The function f‘(x) defined by:

is called the derivative of f with respect to x – or the gradient function of f(x).

Using the definition to find the gradient function (or derivative) of f, is sometimes referred to as “…finding the gradient/derivative using First Principles”

Let’s use it to find some other gradient functions….