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Hello.

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Welcome back.

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In this lesson we are going to give a short revision of calculus.

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This will help you to be able to picture back propagation and gradient descent more easily in future

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lessons.

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You do not need to understand calculus to work with deep learning.

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In fact out say majority of the people working with deep learning nowadays don't even understand the

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calculus behind the back propagation and gradient descent apart from the hard core researchers.

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Of course so let's begin

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this function over here says F of equals 3 A.

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This means we get a result of the function f of E by multiplying the number three by whatever the value

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of is.

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In this graph.

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Over here the x axis represent a

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the x axis represent a and the y axis represents the values for F of E.

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For example let's say a equals to

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then F A is going to be six.

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Since we multiply 2 by 3 to get F of E as we have over here all right.

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And for example let's say we increase a by a tiny amount an amount such as zero point zero 0 1 if we

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increase a base viewpoints for a short one and we compute f a.

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This is what we get we get six point church or three since f of E equals three because we are going

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to perform three more to play by two point zero to one right.

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We can realize that whatever it is F of a is three times that value.

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When we plot our value and our correspondent f of a values this is what we get we get this graph over

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here as you can see we've got two here we increase we get two points you sure one we've got six here

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we increase we get six points or two or three close when we increase this when we compute f off two

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points which are one We get six points for short three right we can compute the slope of F of a bi divide

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into vertical change by the horizontal change over here we have named the vertical change height you

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can see height and the horizontal change has been named as with and this is the height zero point zero

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three the width is your points which are 1 when we perform this division we arrive at the answer 3 as

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we have over here and is the same thing we get over here.

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This is often referred to us the F E over d e f of e of a D E and this is how we write it d e f g over

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D E and we get 3.

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Also I should point out that you don't always need to plot a graph in order to compute the functions

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we should define basic rules for finding derivatives later on in the lesson.

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Let's take a look at another example.

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Over here we have a function Y equals X squared right.

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Another way to find the derivative is the change in y over change in x which is the same thing as what

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we spoke of earlier but we can write it as Delta Delta to represent change.

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So delta y over delta x over here we have a number of x values in the corresponding y values from this

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graph.

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If x is 1 given our equation Y equals X squared then Y will still be a y will be 1.

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Also if x is 3 y will be 9.

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If we find a delta x meaning the change in x we get to if we find a Delta why the change in Y.

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We get 8 if we perform delta y over Delta 2 we get for this is that there effective delta y over delta

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x 8 divided by two equals four.

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Right now let's take a look at some routes of derivatives the power to state star to find a derivative

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of a function x rated a power n.

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The answer becomes.

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And times X reached the power end minus one.

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If N equals zero then then the derivative is zero because that they're effective of a constant is equal

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to zero.

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Right.

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If we want to find the derivative of a constant a multiplied by a function f of x we can find a derivative

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of the function f effects and then multiply the answer by the constant a derivative of a function f

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of x is also sometimes written us f prime of X prime.

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Is this apostrophe sine you have up here.

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So that's another way of writing the derivative.

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D derivative of the sum of two functions f of x and f of T is equal to D.

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Derivative of F of X plus the derivative of F of G as we can see shown in these two examples over here.

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Right.

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You can post a video and take a look at the examples we already mentioned the derivative of a constant

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equals zero.

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Right.

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The derivative of a function multiplied by a constant equals the derivative of the function only then

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multiplied by the constant.

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So if we have the function multiplied by a constant and we want to find a derivative we first have to

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find a derivative of the function alone and then it would multiply the answer by the constant.

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The difference rule works the same way.

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Like the sum rule and we've already seen the same rule we set the m the derivative of the sum of two

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functions f of x plus G of X equals that derivative of F of X plus the derivative of G F X and a differential

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works exactly the same way as the sum rule the product to over here says the derivative of the derivative

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of the product of two functions is equal to D derivative of the first function multiplied by the second

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function plus defense function multiplied by the derivative of the second function the derivative of

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sine X equals cause X and the derivative of course X equals minus sine X right.

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And the derivative of each less constant e raised about X is the same thing E ratio part X the chain

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rule is the one group that is heavily applied in back propagation.

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It states that we apply the chain rule by multiplying the derivative of the outside function by the

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inside function.

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It is often expressed as D X equals DFT y dot do Y D X let's say we want to find the derivative of a

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function over here that they're a victim of this function brackets open three x plus two X squared brackets

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closed or squared.

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We first find the derivative of the entire function which gives us 2 x brackets open 3 x plus 2 x squared

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brackets closed and once we've done that we have to find the derivative of the content inside the front

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inside a bracket.

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And that gives us 3 x plus 4 x.

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So first we find the derivative of the entire function.

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And then we've multiplied by the derivative of the constant in the brackets.

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Only remember the entire function is bracket open three x plus two X squared brackets close squared.

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That is the entire function.

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The content of the frac the content of the bracket only is three x plus two X squared.

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So entire function derivative multiply by content of bracket derivative.

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That is the chain rule right.