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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)).

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differentiation

Differentiation

Newton

An Introduction

Leibniz

slide2

. 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 ….

slide4

As h approaches 0 then 2x + h approaches the gradient of the tangent.

. 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

slide5

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

slide6

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

Gradient =6

X=3

slide8

A general rule:

If f(x)=xn

Then f’(x)=nxn-1

Example:

f(x) = x4

f(x) = 4x3

slide9

More Rules …

If f(x)=axn

Then f’(x)=naxn-1

If f(x)=axn +bxm

Then f’(x)=naxn-1 + mbxm-1

slide10

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

slide11

Leibniz Notation

Leibniz

The derivative is sometime written using a notation introduced by Leibniz

Example:

y = x4

= 4x3

slide12

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

slide13

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

slide14

Equation of Tangent

y=f(x)

y

.P(a , b)

Therefore the equation of the gradient at P

X

slide15

Equation of Normal

The normal is the line perpendicular to the tangent

y=f(x)

y

Remember:

Gradient of tangent x gradient of normal =-1

.P(a , b)

X

slide16

Second Derivative

The second derivative measures the slope of the first derivative. We write it like this:

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