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Calculus for Machine Learning and Data Science
26 hours 57 mins
So far, you've learned derivatives of some simple functions. In order to find the derivatives of more complicated functions, what you have to use is certain rules in order to piggyback from these simple ones into more complicated calculations. The rules you're going to learn next are multiplication by scalars, sum rule, product rule, and the chain rule for compositions of functions. So the first rule is multiplication by scalars. Let's say you have a function f and it's four times a function g. If you want to take the derivative of f, that means it's just going to be four times the derivative of g. And that works for every value of four. So if it's actually a constant c, it can be any constant, then if f is equal to c times g, then f prime is equal to c times g prime. Let's take a look at why this happens. So consider the function y equals x squared drawn over here. And let's take twice that function. So 2x squared, therefore, the function gets multiplied by 2. So let's look at some points. On the left, you have the point 1, 1, and the point 2, 4, because 2 squared is 4. And so the slope of this secant line, well, it's rise over run, where the run is 1, which is 2 minus 1, and the rise is 3, because that's 4 minus 1. So this slope, it's 3 over 1, which is 3. Now let's look at the corresponding points in the function in the right, which as we said, it's double the function in the left. The point corresponding to 1, 1 is 1, 2, and the point corresponding to 2, 4 is 2, 8. And let's calculate the slope. Well, the run is the same as the run in the previous one, which is 2 minus 1 equals 1. But the rise is double the previous one, because in the plot on the right, everything gets doubled in the vertical direction. So the 4 minus 1 equals 3 on the left, which is the rise, becomes an 8 minus 2 equals 6 on the right. And so the slope multiplies by 2, because the numerator multiplies by 2 and the denominator stays the same. Therefore, if the function multiplies by 2, all these slopes get multiplied by 2. Now if you take the limit, if you let the point on the right move towards the point on the left, you still have that all the slopes multiply by 2. Therefore, the derivative or the slope of the tangent, which over here is 2 as we saw before, now becomes 4. In other words, if we take a function and multiply it by 2, the derivative gets multiplied by 2. And this happens for every value. So for example, if we have a function like this, and we multiply it by c, which means the height on the right is c times the height on the left at every point, then the derivative gets multiplied by c. So if we have a point over here, x comma y, with slope f prime of x for the tangent line, on the right, the point corresponding to that is going to be x cy, and the slope is going to be c times f prime of x. Because if you think about it, the plot on the right is obtained from the plot on the left by basically putting your hands on top and bottom, and stretching it by a factor of c. So that is the multiplication by scalar rule.