0:00:06.420,0:00:09.156
Section 8.1 introduction to

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Differentiation. In this[br]section we introduced the

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notion of Differentiation.

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

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

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For example, when Y equals X[br]squared, the gradient of the

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graph is given by two X.

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

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calculate the gradient of Y[br]equals X squared at different

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points on the graph.

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

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

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

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of 6 units for every one unit[br]increase in X.

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Similarly, a gradient of[br]minus 4 means that the

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values of why are[br]decreasing as a rate of 4

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units for every one unit[br]increase in X?

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

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

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

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

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a negative gradient.

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

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Nor does that knowledge after[br]gradient function allows us to

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predict what the graph will look

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

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Let's look at another example.

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When Y equals X cubed.

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The gradient function is 3 X

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squared. Suppose we want to[br]calculate the gradient of the

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graph Y equals X cubed. For AX

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equals 2. BX equals minus one[br]and CX equals 0.

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

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squared, which equals 12.

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

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3 times minus one squared,[br]which equals tree.

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For part C, where X equals 0.

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

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We will now look[br]at some notation.

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

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We write its gradient function[br]as dyd X.

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

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The process of finding dyd X is[br]called differentiation with

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respect to X.

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We will now look at another[br]important example.

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

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the power of N minus one.

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We write if Y equals XDN.

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

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This example gives rise to an[br]important formula and

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therefore should be memorized.

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We will now look at more

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

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

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

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This is also known as the[br]rate of change function.

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You're encouraged to trade[br]exercises to help you

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clarify this material.