The Second Derivative

Hello, I'm Samuel Siksik, professor of math at the University of Warwick.

This is the third course in a series of courses on differentiation.

In previous courses, we introduced the derivative Dy by the X

and explained its relationship to the gradient of

the glass of why Plotted against ex

in this first video of the third course will go to introduce the second derivative.

So sometimes if I start off with the function Y equals F of X.

Sometimes I call the deliberative d Y by D X, which I also like the ff dash of X.

I call this the first deliberative to distinguish it from the second derivative.

So the second deliberative

is what you obtained by differentiating f of X twice,

and we normally write it as d two y by the X squared or F double dashed of X.

So to let's start out with a very concrete example,

given wise equal to X cube plus three x minus one.

To calculate the first derivative dy by the ex, I just differentiate

and I get three X squared minus three.

That's the first derivative for the second derivative.

I differentiate again, so I get d two y by D. X squared is

six x

to just fix ideas very quickly. Here's, uh, an exercise.

So please pause the video and determine the second derivatives

of these three functions. Then we are going to come back and go through them.

Okay, let's take them in terms. So the first function is a sign of X.

And you should know by now that if you differentiate sine of x, you get cause of X.

If you differentiate cause of X, you get minus sine of X.

So the first derivative of y equal sign of X is D Y by the X is equal to cause of X.

The second derivative is going to be d two y by d. X Squared is minus sine of X.

Let's look at the second function.

So the second function is y is equal to e to the two x So we know

the derivative of E to the K X is K times E to the K X.

So the first derivative is going to be dy by the x two e two, the two X

and now I differentiate again.

So I'm going to get four e to the

two X

now the third part. It's a little bit more advanced,

and we need to use the product rule to calculate the first derivative.

So the product rule says that if why is a product f of x times g f x

then d y by the X is equal to f dashed

of x times g f x

plus f of x times G dashed of X.

Okay, so let's compute the first deliberative.

My function is y is equal to x squared times sine of x I let

f of x b x squared I let g of x b sine of x

I know how to differentiate X squared so f dashed of X is two X

I know how to differentiate sine of x g Dashed of X is cause of X

and I just substitute this in the product too.

So I get d Y by the X is going to be two x times sine x plus X squared times cause X

Now I'm not done. All I've done so far is calculated the first derivative.

I want to calculate the second elevator.

So I want to take this function that I've got,

which is to excite X plus X squared cause X

and differentiated.

And because I have a some

I'm going to differentiate each one separately and add the results.

So I'm going to differentiate two x i x.

I'm going to differentiate X squared Kazaks, and I'm going to add the results.

And that will give me my second derivative now

two x sine X to differentiate that again,

I need the product tool.

So in my head, I'm now because we're trying

to do these, uh, more quickly. Okay, we've we've learned how to use the product.

So in my head I'm thinking

two x sine x The f of X is going to be the two x

The g X is going to be the sign X.

If I differentiate that well,

the derivative of two X is to the derivative of sine X is cause X.

So if I differentiate two x sine x using the product tool,

I get to x Kazaks plus to sign X.

Now let's differentiate X squared Kazaks again in my head.

I'm applying the product tool where I take f of x to be X squared g f X to

be cause of X derivative of X squared is two x derivative of Kazaks is minus sine x.

So

I get applying the product tool X squared times minus sine of X plus two X kazaks.

Now I add these. Okay, so I collect terms.

So I'm getting two minus x squared times sine X plus four x kazaks.

Okay,

Make sure that you go through this example

and understand it very well.

In this video,

we introduced the second derivative as the deliberative of the deliberative

and we saw several examples where we computed the second derivative.

In the next video,

we will try to understand what the second

derivative tells us about the graph of a function

and what it also tells us about the station. Lee points of a function