In this video, we formally define the derivative as a limit in two different but equivalent ways and practice these definitions on the cubing function that takes x to x cubed. Recall from earlier videos, that we've looked at the notion of an average rate of change of a function over some interval of inputs, which is just the slope of the line joining the end points of the curve. Recall for displacement function, as a function of time. This produces the average velocity for the trip. If we imagine the time interval becoming shorter and shorter, in fact as vanishingly small, recall that we get the instantaneous velocity, which is the idea behind what you see on your speedometer. This is represented by the slope of the tangent line to the curve at the point of interest. Recall that we've seen this idea informally in an earlier video by constructing slopes of secants to curves and watching what happens as they get shorter and shorter. As the secants vanish away, the slopes of the secants get closer and closer to the slope of the tangent line. We think of them as reaching the slope of the tangent line in the limit and we express this in limit notation. We take this now as the definition of the derivative of the function f given by the rule y equals f of x at the input x. The derivative of f at x denoted by f dashed of x is a limit if it exists as h tends to zero of the quotient f of x plus h minus f of x divided by h. This represents the slope of the tangent line to the curve y equals f of x at the point with coordinates x, f of x. In the case of f of x equals x squared, the squaring function, recall that we calculated this limit in an earlier video, but the derivative is 2x. So, f dashed x is two times x. Let's move from squares to cubes and consider the function y equals f of x that takes an input x to execute and think about what the derivative might be. The graph of this function looks like this, with a 180-degree rotational symmetry about the origin, it turns out to be an odd function, an important concept we'll discuss in detail later. Apart from being a lot steeper, this curve has an important distinguishing feature from the parabola that describes the squaring function. If you draw miniature tangent lines to the curve throughout, you'll discover that the tangent to the curve at the origin, which is just the x-axis in this case, crosses the curve. The origin becomes an inflection point for this curve. Another important concept we'll discuss in detail in a later video. Already you can see for that particular tangent line, the slope is zero since it's just the x-axis in fact which is horizontal. But what about slopes of tangent lines in general for f of x equals x cubed? Well, let's work through the definition of the derivative and say what happens. Start off with the definition. F dash of x is the limit as h goes to zero of f of x plus h minus f of x over h, which in our case becomes the limit, as h goes to zero, at x plus h cubed minus x cubed over h. The cube and the numerator expands, you can check as x cubed plus 3x squared times h plus 3x times h squared plus h cubed minus x cubed. We noticed the x cubed and minus x cubed cancel out. So, we get the limit as h goes to zero and 3x squared times h plus 3x times h squared plus h cubed all of h, and the numerator factor arises as 3x squared plus 3x times h plus h squared times h. We can cancel the h's in the numerator and denominator, and the whole thing simplifies the limit as h goes to zero as 3x squared plus 3xh plus h squared. There's no problem simply evaluating this expression for h equals zero, reducing quickly to 3x squared. This solves our original problem. The derivative of f of x equals x cubed is 3x squared. Recapping from an earlier video with this new notation, we've already seen that if f of x is x squared and the derivative is 2x, and just now, that if f of x is x cubed, then the derivative is 3x squared. You can easily work through the details for f of x equals x to the fourth, though it takes a bit longer and find that the derivative is 4x cubed. This all instances of the general pattern. For f of x equals x to the n, for any exponent n, the derivative is n times x to the n minus one. That is, the exponent falls down to the front and you get a new exponent by subtracting one. If you know about something called a binomial expansion, you can verify this directly already for any positive integer n from the definition of the derivative. When we've developed some general techniques with finding derivatives, this result will follow quite easily and works for any real expanded n, which is very nice indeed. There's an alternative limit definition of the derivatives that comes about naturally by taking the average rate of change of the function over an interval and seeing what happens in the limit as the interval shrinks to nothing at the left-hand end point. Here's our typical curve y equals f of x. To find at least an interval from A to B, producing outputs f of a and f of b for the respective endpoints, and then we draw the secant joining the end points which has slope f of b minus f of a over b minus a, and this is just the average rate of change. If we allow b to head towards a, then the secant lines up more and more in the direction of the tangent line to the curve at x equals a. So, that its slope the, average rate of change approaches the slope of the tangent line, which is just the derivative of the function of x equals a, denoted by f dashed of a. Thus, we expect an alternative definition of the derivative namely, f dashed of a, equals the limit as b approaches a, and f of b minus f of a over b minus a. Now, this looks quite different to our original definition. So, let's see if we can recover the original definition by rearranging this formula somehow. Put h equal to b minus a, because we want to get h in the denominator, and put x equal to a because we want x in the game. Observe that b is just a plus h, which is x plus h since x equals a. And h, the difference between a and b, tends to zero as b gets closer and closer to a. So, the derivative of f at x which is called f dashed of x is just f dashed of a, since x equals a, which feeding in our new alternative definition in the box above is the limit as b approaches a of f of b minus f of a over b minus a, but b is x plus h, a is x, b minus a is h, and b tends to a as h tends to zero. So, we can rewrite this as the limit as h tends to zero of f of x plus h minus f of x all over h. We recover our original definition of the derivative that we introduced early in this video. So, everything is working nicely as expected. Let's revisit the derivative of the cubing function but using this alternative definition. The definition is in terms of f dashed of a and says that we take the limit as b approaches a of f of b minus f of a divided by b minus a. But f of b is b cubed, and f of a is a cubed. Now, there's a beautiful factorization of the difference of two cubes that you can check if you've not seen it before. b cubed minus a cubed is b minus a times b squared plus ba plus a squared. So, we get cancellation of the factor b minus a in the numerator with b minus a in the denominator, so that it reduces to finding the limit as b goes to a of b squared plus ba plus a squared, which becomes straightforward evaluation replacing a by b to get a squared plus a squared plus a squared which is 3a squared. This shows that f dashed of a equals 3a squared. If we revert to using x instead of a, we get f dashed of x is 3x squared. Exactly what we obtained using the original definition which is very pleasing. Today, we see that many ideas we've encountered previously are all coming together using the unifying concept of a limit. We define the derivative to be the slope of the tangent line to a curve which has two quite natural and equivalent formulae in terms of limits. We use both of these formulae to find the derivative of the function that takes x to x cubed and the answer turns out to be 3x squared. In later videos, we'll apply this limit formulae for the derivative to other important functions. Then in the next module, develop some general techniques and principles that enable one to find swaths of derivatives easily and quickly in a variety of different settings. Please read the notes and when you're ready please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
