www.mathcentre.ac.uk/.../81-introduction-to-differentiation-podcast.mp4

Section 8.1 introduction to

Differentiation. In this
section we introduced the

notion of Differentiation.

Given a function for example Y
equals X squared, it is possible

to derive a formula for the
gradient or slow plus the graph.

For example, when Y equals X
squared, the gradient of the

graph is given by two X.

The important thing to note is
that using the formula we can

calculate the gradient of Y
equals X squared at different

points on the graph.

For example, at X equals 3, the
slope or gradient is 6.

Or when X equals minus two, the
slope or gradient is minus 4.

A gradient of 6 means that the Y
values are increasing at a rate

of 6 units for every one unit
increase in X.

Similarly, a gradient of
minus 4 means that the

values of why are
decreasing as a rate of 4

units for every one unit
increase in X?

Noah stop and X equals 0 the
gradient or slope is 0.

On the handout is a graph of the
function Y equals X squared.

Study into graph. We see that
One X equals tree. The graph has

a positive gradient. And when X
equals minus two, the graph has

a negative gradient.

Also, when X equals 0, the
gradient of the graph is 0.

Nor does that knowledge after
gradient function allows us to

predict what the graph will look

like. In other words, is the
graph increasing or decreasing?

Let's look at another example.

When Y equals X cubed.

The gradient function is 3 X

squared. Suppose we want to
calculate the gradient of the

graph Y equals X cubed. For AX

equals 2. BX equals minus one
and CX equals 0.

For party for X equals 2, the
gradient or slope is 3 * 2

squared, which equals 12.

For party for X equals minus
one the gradient or slope is

3 times minus one squared,
which equals tree.

For part C, where X equals 0.

The gradient are slope is 3 * 0
squared, which equals 0.

We will now look
at some notation.

If Y is a function of X, that
is, if Y equals F of X.

We write its gradient function
as dyd X.

This is not a fraction, even
though it might look like 1.

The process of finding dyd X is
called differentiation with

respect to X.

We will now look at another
important example.

For any value of N, the gradient
function of XTN is N times X to

the power of N minus one.

We write if Y equals XDN.

Then dyd X equals N times X to
the power of N minus one.

This example gives rise to an
important formula and

therefore should be memorized.

We will now look at more

notation terminology. When Y
equals F of X, alternative ways

of writing the gradient function
dyd X ry dashed or why primed?

At the relative is another name
given to a gradient function.

This is also known as the
rate of change function.

You're encouraged to trade
exercises to help you

clarify this material.