Differentiation

1 / 16

# Differentiation - PowerPoint PPT Presentation

Differentiation. Newton. An Introduction. Leibniz. . Q. Find the gradient of the curve f(x)=x 2 at the point P (x, f(x)). f(x +h). . P. Draw a tangent at P and try and find the gradient of this tangent. f(x). x. x + h. Approximate this tangent by a chord PQ where Q=(x+h,f(x+h)).

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Differentiation' - metta

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Differentiation

Newton

An Introduction

Leibniz

. Q

Find the gradient of the curve f(x)=x2

at the point P(x, f(x))

f(x +h)

. P

Draw a tangent at P and try and find the gradient of this tangent.

f(x)

x

x + h

Approximate this tangent by a chord PQ where Q=(x+h,f(x+h))

Now we can find the gradient of this chord ….

. Q

But as h approaches 0 we can see that 2x + h approaches 2x

. P

We say that 2x is the limit of 2x+ h as h approaches 0 and write this as:

h

We now know the slope of the x2 at any point is 2x

We call this gradient function the derived function or derivative

We denote the derivative of f(x) by f’(x)

The process of finding the derivative is called differentiation

To summarise:

If f(x)=x2

Then f’(x)=2x

This means that if at the point x=3 the slope of the curve is 2(3)=6

X=3

A general rule:

If f(x)=xn

Then f’(x)=nxn-1

Example:

f(x) = x4

f(x) = 4x3

More Rules …

If f(x)=axn

Then f’(x)=naxn-1

If f(x)=axn +bxm

Then f’(x)=naxn-1 + mbxm-1

Examples

f(x)=2x3 +4x2

f’(x)=6x2 +8x

f(x)=3x-2 - x-3

f’(x)= -6x-3 +3x-4

f(x)= x-1/2

f’(x)= -½x-3/2

f(x)= 4

f’(x)= 0

Leibniz Notation

Leibniz

The derivative is sometime written using a notation introduced by Leibniz

Example:

y = x4

= 4x3

Rate of Change

The derivative measures the slope of a curve. When the curve represents a relation between variable x and a dependant variable y we interpret the derivative as the rate of change of y with respect to x

y

Gradient =Rate of Change of y with respect to x

X

Equation of a Line (Revision)

We know the equation of a line is :

If P(a , b) is a point on the line then we can find C

Equation of Tangent

y=f(x)

y

.P(a , b)

Therefore the equation of the gradient at P

X

Equation of Normal

The normal is the line perpendicular to the tangent

y=f(x)

y

Remember: