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Okay so we've generated the data and we've plotted a 3D chart.
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Now before we run our gradient descent algorithm, we're going to have to calculate the slope or the gradient
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on all the different points on our chart.
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We're going to have that down, because remember the gradient was what told us how far away we were from
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the minimum, the steeper the gradient the further away we were and the higher our cost in our wonderful
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three step machine learning process. Now previously our cost functions were a bit more simple, right?
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All we had to do was apply our power rule and work out our derivative function.
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But for our cost function in example 4m that's not going to be enough.
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And the reason for this is is that this function as a little bit more complex. Before we only had a slope
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in one dimension - the x dimension.
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But this time we've got a slope in two dimensions because we've got two parameters -
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we've got the X and the Y. And not only that, our functional form is a little bit more complex as well.
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We've got a fraction, we've got exponents, it's getting a little bit more complex to do that derivative
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here, it's
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not such a simple process.
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And this is a little bit of a preparation to the kind of pretty hairy cost functions that you might
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encounter
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going forward. A lot of those come from statistical theory and there are definitely a lot more involved.
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So what I want to show you in this lesson is how you can tackle quite complex cost functions without
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having to work everything out by hand.
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So as I said before, this time around we've got a Y and we've got an X, and we're optimizing our cost
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across two parameters. Just because the Y is low doesn't mean that the cost is low.
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Right?
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And this makes sense if you think about it. Looking at this graph just because we've got a low Y doesn't
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mean that the X is low as well.
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We have to pick both an optimal X and an optimal Y.
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And this means working out two slopes - a slope for each parameter. And to calculate the slope we are going to calculate
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the partial derivative.
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Now, what's a partial derivative?
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It's taking a derivative of a function with respect to one of the variables and holding the other
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one constant.
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So if we're taking the partial derivative with respect to X then we're holding Y constant.
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That way we get the gradient in the X direction and the way we're going to go about this is I'm going
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to show you how to calculate a partial derivative using symbolic computation in Python.
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So back in our Python notebook at the bottom let's add a couple of cells and then the very first one
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we're going to change to Markdown and we're gonna add a section heading here.
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This section heading is going to read "Partial Derivatives & Symbolic Computation".
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Now the question is what does a partial derivative actually look like and how do you work it out? Well,
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if you remember your high school maths then this isn't gonna be a problem for you.
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You're gonna be looking at this function right here and you're already going to be writing out the partial
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derivatives for it.
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But on the other hand if you're a bit rusty with your calculus or you simply want to check your work,
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let me show you some alternative ways that you can solve this mathematical problem.
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The first approach is being a little sneaky and typing the words "derivative calculator" into Google to
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find an online derivative calculator that will solve this for you. I have to say this is actually a pretty
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good idea.
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So there's a couple of really good ones out there.
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You know, you'll get a lot of results.
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I checked a couple of them out, the one I quite liked was symbol lab.
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So this is a pretty decent one.
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So on every single one of these Web sites, you're gonna get some text box here where you have to enter your
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equation.
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So ours was 1 divided by 3 and then it was to the power of "-x^2 -
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y^2"
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So you're gonna be typing this in and it's going to be "1/(3^(-x^2
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-y^2))" and then you got to make sure on this particular website that you go back down
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outside of the exponent and then put "+1" here.
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Otherwise you'll get a very, very different answer than if you have it up top. When you hit "Go", it calculates
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the derivative for you.
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So we've got our solution.
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So right here you're looking at the partial derivative with respect to x. And the really cool thing about
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a lot of these Web sites is that you can actually see the steps on how they got the solution.
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So you can see them applying the exponent rule, applying the chain rule and then down here
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they're applying our old friend the power rule and they're also showing you how to simplify the solution
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in the end. Now checking out a Web site like this,
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and you know kind of, checking the solution or revising your calculus is all very well and good but you
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know, you and I we're doing a Python course together.
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So how do you do this in Python and in Jupyter notebook?
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And this is what I'm going to show you right now, because we're going to be doing symbolic mathematics
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directly in Python using a module called "SymPy".
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Sympy actually has their own Web site as well so you can check out the documentation here if you like.
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You can see who's developing it.
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You can even donate money to them.
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So,
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so you know these guys are doing good work, it's a good project.
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I think the design is a little bit dated, needs a bit of a facelift but hey they're doing symbolic mathematics,
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not designing for Apple or what have you.
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Now as always when we want to use an external module in our Jupyter notebook, then we're going to have
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to import it and we're going to import it at the top where we're importing all the rest of our modules.
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So keeping all our imports together nice and tidy.
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So from SymPy we're interested in two things,
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we're gonna say "from sympy", all lowercase, "import symbols, diff".
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So we're interested in importing two things.
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One of them is the symbols functionality which will help us write mathematical notation.
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And the other is diff, which allows us to differentiate a mathematical function.
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All right.
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So let me scroll back down and show you how SymPy works.
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Also let me fix this typo here.
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First off, we need to tell Python which mathematical symbols we're gonna use.
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So instead of X and Y I'm going to use a and b to show the difference between our Python code a bit more
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clearly.
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So I'm going to say "a, b = symbols(
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'x', 'y')"
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By writing this line of code or telling Python that a stands for x and b should stand for y in the code
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that we're gonna be writing next.
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Mm hmm.
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Now I realize this sounds a little bit confusing.
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So I'm actually going to print out our function as it is below our cell using this notation. So I'm gonna
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say "f(a,b)" and hit Shift+Enter. Now if you see the error "symbols is not defined",
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then you've got to hit Shift+Enter on our import cell. What you should see instead is this,
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you should see that Python now recognizes our function as a mathematical function and it's substituted
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x and y for our a and b
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arguments. So we can see here that thanks to SymPy, a and b are now treated as the x and y in our function
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and Jupyter notebook now understands how to work with this function as we would do with a mathematical
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equation.
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So let's work out the partial derivatives using the diff function. So let me wrap this in a print statement
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I'm going to say print and then single quotes
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and in those single quotes I'm gonna say "Our cost function
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f(x, y) is: ", and then that's gonna be the end of the string comma, "f(a,b)".
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So I'm going to this in a print statement and I'm going to add some more print statements below because
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we can now work out the partial derivatives with this diff functionality from SymPy. Because we've imported
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it above, it's very, very simple to call.
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It's "diff(f(a,b))" and then you have to provide a second argument to this.
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You have to say what you're differentiating with respect to.
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So if we wanted to differentiate with respect to our first parameter, then we would differentiate with
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respect to a. Hitting Shift+Enter,
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we see that this here is the same partial derivative that we got from using one of the online derivative
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calculators.
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Let me put that in a print statement as well.
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So I'm going to say print, single quotes, comma and in those single quotes I'm going to say "Partial derivative with
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respect to x is: " and then have the output that we just had here and then close the parentheses at the
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end and hit Shift+Enter.
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So now we know how to work with our cost function and how to work with our partial derivatives and this
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brings us to our next point.
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How do we calculate the cost at a particular point in our function? For that we need to evaluate our
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function.
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So we need to substitute the particular x and y values, right, like 1 and 1.5 or what have
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you into our function.
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The SymPy module gives us this ability through another function, namely evalf - evaluate function.
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So here's how you would use it on our cost function.
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I would have "f(a,b).evalf", and then parentheses and in those parentheses I can substitute
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the values at which point I want to evaluate the cost.
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So I would say "subs = {a:
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1.8, b:1.0}".
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Then close the curly braces, close the parentheses for the evaluation function and then hit Shift+Enter.
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And what you should see is that the cost at this particular point is approximately 0.99.
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If I take a look at our graph, I take a look at where x=1.8 and y=1 is, it's around here
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and going up I'm at around 0.99 on the z axis.
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Okay so that's pretty useful.
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The only thing that's a little bit new is this notation right here.
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This is Python code that we haven't quite seen before, because we've got these curly braces
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and then inside we have two values - a and b, separated by a comma and then we set the values with colons.
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So a gets the value 1.8 and b gets the value 1.0.
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So this notation with the curly braces
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together with the colons is the Python notation for giving both a and b a value at the same time.
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The specific term for this is a Python dictionary.
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We've got a key value pair.
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A is the key and 1.8 is the value.
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And then we've got another key value pair, b is the key and 1.0 is the value.
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Now if you're coming from another programming language, a dictionary might have been referred to as an
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associative array and it's very similar to an array really but instead of kind of grabbing a particular
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value by an index, by a number, as we have done with our array, say getting the first one or getting the
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second one,
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in this case we are working with a key,
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so it's like our index has a name.
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And the reason they're called dictionaries is because they pretty much work like real dictionaries.
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If you have a word say "bug", then that's the key.
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And if you have the definition of this word - "unintended behavior or defect in a program that causes
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it to crash or malfunction", then that's the value and that's it.
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That's the dictionary data structure in a nutshell.
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You can always spot it with the curly braces and the keys separated from the values by a colon.
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All right.
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So now that I've printed out the cost for our cost function at a particular point, I'm going to wrap this in
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a print statement again, I'm going to say "print('',)", and in the single quotes I'm going
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to write "Value of f(x,y) at x=1.8, y=1.0 is: " and then I'll have
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the rest of our print statement like this and I'll finish it all off with a parentheses at the end. I'm going to
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split it out over two lines like this,
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so it's a bit more easy to read.
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I'm going to hit Shift+Enter and evaluate this cell again. Now as a really, really quick practice,
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I want to give you a challenge, namely can you evaluate the value of the slope with respect to x at this
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very, very same point, at 1.8 and 1?
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And can you print out this value, the value of the partial derivative with respect to a at 1.8
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and 1 in the cell?
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I'll give you a few seconds to pause the video and figure this out.
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All right.
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So here's the solution.
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Once again you're going to be working with the evalf functionality, but in this case we're not gonna
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be evaluating the cost function, we're gonna be evaluating the partial derivative, so we can say 
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"diff(f(a,b),)" and then we have to say if we want to evaluate this function
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with respect to a or to b. I'm going to say a and then I'm going to close the parentheses for our diff.
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So this is our partial derivative.
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We'll put a dot after it and then again we'll say "evalf(subs=)" and now we'll provide
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our dictionary, open curly braces "a:1.8, b:1.0" and let's print
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this out with Shift+Enter and here we can see that the slope at this particular point in the x direction
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is 0.037. Once again, I'm going to wrap this in a print statement and I'm going to say "Value of partial
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derivative with respect to x: ", and I'm going to close the parentheses at the end and hit Shift+Enter.