0
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Now suppose we want to find the value of x that minimizes the cost - that minimizes f(x). So on the
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graph,
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we can already see that that point would be around here.
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But let's think about this problem a bit more deeply.
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Right?
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So if X was equal to 2, if we were here on the graph, then the cost would be decreasing a lot as we're
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moving down.
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But when X is equal to 1, then that costs still decreasing,
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but by not as much because the line just isn't as steep.
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Right?
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And by the same token when X is equal to 0,
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that line is like flattening out even more.
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So at this point the cost is only decreasing very, very slightly.
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Then we'd reach our minimum and then the cost starts increasing again,
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right, as we move up the curve.
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So in other words, the slope or the steepness of this function,
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yeah, is telling us when we've reached our minimum cost because that slope is the rate of change.
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And when this slope stops changing,
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well, on this graph we know that the cost is equal to the lowest value.
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In other words the slope of this function is kind of equal to 0 at the point where the cost is the
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lowest.
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Now you may or may not remember from school how to find the slope of a function,
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depends how long it's been, right?
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But, as a reminder, the slope of a function is given by that functions derivative.
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So let's add this section heading, down in the next cell.
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I'm going to go to "Cell" > "Cell Type" > "Markdown", put two hash tags there, and then write "Slope & Derivatives"
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because this is what this lesson is gonna be about. Now as a challenge,
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can you create a Python function for the derivative of f(x) and call that function df(x).
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I'm going to scroll back up for a second here.
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So remember f(x) was equal to x^2+x+1.
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I'll give you a few seconds to figure this out.
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All right so here's the solution.
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Yeah, if you were scratching your head at this point because your calculus is a little bit rusty, let
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me show you what the solution is and also how we arrived at it.
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So I'm going to define a function with the def keyword, then going to call it "df" and have one input, namely
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x, colon, new line, "return 2*x+1", and that's it.
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So how did I arrive at this?
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How did I know it was gonna be 2x+1? So the answer is I used calculus. Now calculus and derivatives
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are a pretty big topic but there's a couple of mathematical tricks or shortcuts that we can use to calculate
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derivatives very, very quickly and one of these tricks is called the Power Rule and we'll be applying
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it in several of the upcoming examples.
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So here's a very quick reminder on how it works, if you have a function x^n, then the
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derivative of this function is n * x ^ (n-1)
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So that's a pretty simple rule, right?
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Easy to remember. But before we move on let me give you a couple of quick examples.
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So, say we have a cost function that is x^5 and we want to calculate this cost function's
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derivative.
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If we apply the power rule then we simply get 5*x^(5-1), which is 
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5*x^4.
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And that's all there is to it.
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That's the function of the derivative.
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And here's another very, very quick example - we can apply the power rule to this function as well.
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x^3+5*x^2+7.
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So I'm going to go through these terms one by one - x^3 is pretty easy right?
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Because x^3 will simply be 3*x(3-1) and then for that second
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term, 5x^2 the five stays where it is and we have 2*x^2-1.
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And then that 7 right - that 7 just disappears.
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That 7 just drops because 7 is a constant.
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It doesn't depend on X - constants don't change, and since a derivative is all about the rate of change
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the constant drops out.
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It's equal to 0.
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So the 7 disappears.
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So what we end up with is 3*x^2
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+ 5*2*x, right, 10*x, and that's it.
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Right.
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That's the power rule for derivatives in a nutshell. So back in Jupyter notebook,
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say we want a graph our derivative function that we've just worked out here.
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We've worked hard to create it.
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So let's visualize it side by side with our original cost function.
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So what we want is we want to charts, side by side, and to do that we're going to have to use the so-called
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subplot functionality from matplotlib.
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So I'm gonna show you the subplot functionality and I'm also gonna show you how to size the two charts.
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So this is something new.
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This is something we haven't done before, but since a lot of the code will be very, very similar to what
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we've had up here I'm just gonna very, very quickly go to "Edit" > "Copy Cell" and then go down here
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and I'm going to go to "Edit" > "Paste Cell Above", so I can reuse some of this code that we've got here. And
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I'm going to change my comment here and I say "Plot function and derivative side by side".
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So this is what we're gonna do here, and uh, in order to size our plot,
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yeah,
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in order to size it we're gonna have to set the figure size of our plot.
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So we're gonna write "plt.figure()", and then we're gonna say "figsize="
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and then we have to give it a width and height.
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So let's see what happens if I put in, I don't know, "10,5" in the square brackets and hit Shift+
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Enter, so you can see now that the width is 10 and the height is 5.
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I looked at the documentation for this and this is actually measured in inches of all things.
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I don't really know.
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I don't really know why that is but I hope it makes our American friends happy.
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Now I'm going to modify this code a little bit more, so in order to have two plots side by side we're
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gonna have to use this subplot functionality from matplotlib, so I'm going to use "plt.subplot" and then
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I have to specify three numbers - a row, a column and an index.
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So in this case I'm going to have two plots, side by side.
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So they're going to be in the same row.
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So one row is going to be two columns.
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And the first plot, this cost function here is gonna be, at index 1, is gonna be my first plot. And now
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it's time to add our second plot.
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So, I'm going to say "plt.subplot"
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And for the second plot it's gonna be still in the same row - row number one, still two columns, but this
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one is gonna be at index 2.
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So this is gonna be my second plot.
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Now I haven't added it yet, I'm going to do that
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now. To graph our derivative, I'm going to write "plt.plot" and we're going to feed in our 
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x_1 values and then we're gonna call our function.
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Right.
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So the first one was f(x_1) and this one is gonna be df(x_1). Let's see what happens
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now. I'm going to press Shift+Enter.
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Huh.
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So I've got two plots
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now. You see this? Two plots side by side and they're still occupying the same space as before, right.
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The overall figure is still 10 by 5.
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So we can see here the figure size hasn't changed, but now it's split with two plots, two subplots taking
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up that same space.
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So I can probably make use of this space here on the right a little bit better by maybe making this
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15 by 5.
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I think that should be about right.
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Press Shift+Enter.
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I get it like this.
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Very nice.
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Now what I'm going to do is add a few labels and style these graphs a little bit better so that if you
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ever come back to this notebook in the future, you can make sense of these charts very quickly and very
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easily.
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So, I'm going to insert two comments first, right.
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So this is the first chart and this is the cost function and down here is the second chart
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of the derivative.
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For the first chart let's explicitly state that it should be blue and that it should have a line width
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of maybe 3 and for our second chart we're gonna give it the color sky blue and a line width of
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5.
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How does that look?
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Pretty good.
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Now let's add some title and X labels and Y labels to that second chart as well.
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So I'm just gonna copy this code here and add it here but instead of having it say "Cost function", I'm
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going to say "Slope of the Cost function" and on the x is still x, but on the y it's gonna be "df(x)"
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and then I'm also gonna set the axes, so again I'm going to copy xlim and ylim and add this down here,
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but I'm gonna have it go from maybe -2 to 3, and on the Y from say, 3 to 6.
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Oh sorry, -3 to 6.
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I'm going to hit Shift+Enter to see what it looks like.
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So this is quite good.
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This is very, very nice and clear but there's one small improvement that I'd really like to make.
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We're interested primarily in where this minimum is here and one of the things that be quite nice to
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see on this chart is where our slope, our sky blue line here, actually intersects the axis here.
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So what we can do is we can style our chart to have a grid.
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This is fairly easy to do so for our second chart for our derivative all we have to do is add "plt.
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grid" and then two parentheses for the method name.
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No need to add any arguments here.
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Hit Shift+Enter and then you can see matplotlib applies a grid automatically to this subplot and now
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it's very, very nice to see that our intersection of the slope of the derivative is going to be probably
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somewhere at the midpoint between -1 and 0, so that's gonna be around here somewhere.
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That seems about right.
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So as a quick summary we've covered a three new things for styling graphs and charts.
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One of them is this idea of sizing them.
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The second thing is this idea of having plots side by side and that requires the subplot function which
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needs to know how many rows, how many columns and what the index is.
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And then also we've used the grid function for the first time to superimpose a grid on to the plot.
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Now as a challenge, can you instead of having these plots side by side, stack them from top to bottom?
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What would you have to change in the Python code in order to achieve this? In order to arrange the plots
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in a different way.
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I'll give you a few seconds to pause the video and try it out. Did you solve it?
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Did you manage to get it?
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Here's the solution.
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The key on arranging the charts is gonna be where our subplots are, right?
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So if you want to have them top to bottom, then you'd have to have two rows and one column and I'd have
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to change this for both subplots.
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So for the first chart and for the second chart I'm going to have to go 2 and 1 instead of 1 and 2. If
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I press Shift+Enter now, we can see our charts arrange themselves from top to bottom. But one of the reasons
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why they look terrible like this is because we've constrained our figure size to be still using the
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same dimensions - 15 by 5, right?
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So it's stretching out the charts across the whole space that it's allotted.
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And that's why they look like this.
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But if we changed our figure size with some new constraints say 5 and 15, right, then pressing Shift+Enter
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will give us the charts like this.
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In this case they're more tall than wide.
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Now I quite like the original setup so I'm going to change this back to 15 and 5 and I'm going to go
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with one row, two columns for my subplots - one row and two columns for both of them.
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So this is how I would prefer to look at these two charts.
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But yeah, if you figure this out super handy for being able to arrange the charts in your Jupyter notebook.
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All right.
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So I hope you're ready for the next lesson.
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I'll see
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you there.
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Take care.