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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


. 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 ….



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


We now know the slope of the x tangent.2 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: tangent.

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



A general rule: tangent.

If f(x)=xn

Then f’(x)=nxn-1

Example:

f(x) = x4

f(x) = 4x3


More Rules … tangent.

If f(x)=axn

Then f’(x)=naxn-1

If f(x)=axn +bxm

Then f’(x)=naxn-1 + mbxm-1


Examples tangent.

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 tangent.

Leibniz

The derivative is sometime written using a notation introduced by Leibniz

Example:

y = x4

= 4x3


Rate of Change tangent.

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) tangent.

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 tangent.

y=f(x)

y

.P(a , b)

Therefore the equation of the gradient at P

X


Equation of Normal tangent.

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


Second Derivative tangent.

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


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