In this video, we use the limit definition to rapidly expand our repertoire of derivatives. The process of taking derivatives of functions is called differentiation. We discussed the additivity of the derivative and how that applies in particular to provide simple rules for differentiating polynomials, explain why the natural exponential function replicates itself under differentiation, and sketch a proof that the derivative of the sine function is the cosine function. We gave a definition of the derivative in terms of limits last time. Let y equal f of x be the rule for a function f. Remember, we defined the derivative of f at x denoted by f dashed of x to be the limit as h tends to zero of f of x plus h minus f of x divided by h. Recall, this limit formula captures precisely the value of the slope of the tangent line to the curve at the point of interest. By now, you're probably starting to picture this phenomenon in your mind's eye and the idea of the limiting slope of secants. We've already applied this formula to discover that if f of x is x squared, then the derivative is 2x, and if f of x is x cubed, then the derivative is 3x squared. More generally, if f x is x to the n, then the derivative is nx to the n minus one. The simplest functions that differentiate quickly are the linear functions with rules of the form f of x equals ax plus b, where a and b are constants. This graphs are just straight lines. If you reflect for a moment, the only reasonable way for a given straight line to be approximated by a tangent line is to take the line itself, and the approximation is so good they match perfectly. In other words, a straight line coincides with this tangent line at every point, and has slope a so that the derivative must be just a. You can check yourself this follows also quickly from the limit definition. So, f of x equals ax plus b implies that f dashed of x equals a. In particular, if a is equal to zero, so that f of x equals b is a constant function, then the derivative is zero. If f of x equals x, then the derivative is one. The derivatives additive by which we mean that if f of x is the sum of two rules, u of x and v of x, then the derivative of f is the sum of the derivatives of u and v. Constants come out the front in the sense that if f of x is k times g of x, where k is a constant, then the derivative of f is just k times the derivative of g. This follow easily from the corresponding properties of limits and details are given in the notes. It's now straightforward to differentiate any polynomial. You differentiate any power of x you see, multiply through by any constant you see, and add everything up. For example, let f of x be this polynomial degree four. Then, the derivative is formed, first by differentiating x to the fourth, which is four x cubed, then move on to the next term involving x cubed. So, you subtract two times 3x squared, then move on to the next term involving x squared. So, you add five times 2x, then move on to the next term involving x, so subtract three times one. Finally, move on to the constant term whose derivative is zero, and add everything up to get this cubic. With some practice, you'll go immediately to this last step. Notice that differentiating a polynomial decreases its degree by one. So, you can always make a polynomial disappear if we differentiate it often enough. For example, suppose f of x is this quadratic, then a derivative is this linear function. If we differentiate again, we get what's called the second derivative, which in this case, the constant six. If we differentiate it again, we get what is called the third derivative which in this case is zero. Differentiating three times makes this quadratic disappear and that's true for any quadratic. If you take a polynomial of degree n and differentiate it n plus one times, then it disappears. Next, we'll look at a function that turns out to be indestructible with respect to differentiation. Consider the function y equals e to the x. Recall from an earlier video, that the tangent line to it's curve at the y intercept has slope one. The base is Euler's number e and is chosen deliberately so that this property holds. We want to transform this fact into a very specific limit. Let's have a look at the curve for x close to zero. If we input x equals h some small positive real number and move up to the curve, then we have the value e to the h on the y axis. Now, draw a secant that joins the curve's y intercept to this point on the curve. The slope of the secant is the vertical rise e to the h minus 1 over the horizontal run h. As h tends to zero, this slope tends to the slope of the tangent line which is one. Thus, the limit as h tends to zero of e to the h minus 1 over h is just one. This innocuous-looking limit is very special, and in fact, the foundation for the central role Euler's number e plays in mathematics. Now, let's prove the following. That f of x equal to e to the x, we claim that the derivative is just e to the x. Here's the proof. First write f dash x using the limit definition. The numerator inside the limit then becomes e to the x plus h minus e to the x, but e to the x plus h can be rewritten as e to the x times e to the h by applying an exponential law. Then, this new numerator factorizes as e to the x outside e to the h minus 1. But e to the x doesn't involve h at all, so it's like a constant that can come out the front of the limit. So, we'll get e to the x times the limit as h tends to zero of e to the h minus 1 over h. But this is the special limit we identify before which evaluates to one. So, the whole expression becomes e to the x times 1 which is just e to the x, and the claim is proved. This is an amazing fact. This function reproduces itself upon differentiation. No matter how many times you differentiate it, the function remains perfectly and obstinately intact. The circular functions y equals sine x and y equals cosine x form a pair of indestructible functions from the point of view of differentiation in an interesting roundabout way as you'll soon see. We claim that the derivative of sine x is cos x. I'm going to sketch a proof. This will be quiet an advanced mathematical argument. If you don't follow all the detail, don't worry. I'll first try to convince you visually that the claim seems plausible. Here's a graph of y equals sine x, and directly beneath it we've drawn the graph of y equals cos x. Here are points on the sine curve, where the tangent lines are horizontal, having slope equal to zero. Notice, the cos x takes the value zero for exactly the same inputs. Let's also focus on points where the tangent lines of y equals sine x, have slope plus one. Notice, the cos x takes the value plus one for exactly the same inputs. Finally, focus on points where the tangent lines y equals sine x, have slope negative one, and notice that cos x takes the value negative one for exactly the same inputs. So, our claim is looking plausible based on these samples of points. Now, we're going to sketch a proof of this claim, and there is more detail in the notes. As preparation for the proof, let's first look more carefully at the tangent to the sine curve at the origin, and ask for its slope. Which looks like it should be one. Take a point on the curve and form the secant that joins the origin to this point. The horizontal run is some number h, say, while the vertical rise becomes sine of h. So, the slope of this secant is sine h of h. So, the slope of the tangent line is just the limit of this expression as h tends to zero. But we calculated this limit in an earlier video using the Squeeze Theorem, with x used instead of h, and the answer is one. This confirms, that the slope of the tangent to the sine curve at the origin is one. Let's also look at the tangent to the cosine curve at x equals zero. Because the cosine curve turns around at that point, the tangent is horizontal. So, must have slope zero. Let's interpret this as a limit. Join the point at the apex on the y-axis at y equals one, to some other point on the curve forming a secant. The new point has coordinates h say on the x-axis, and cos h on the y-axis. The secant is sloping downwards. In moving along this secant, the horizontal run is h, also vertical rise is actually a decline of one minus cos h units. So, cos h minus one it's negative, and if we divide it by h, we get the negative slope of the secant. We have zero being the slope of the tangent line, which is also the limit as h goes to zero of the slopes of the secants. We have that the limit as h goes to zero of cos h minus one of the h is zero. Thus we have two particular limit formulae, one involving sine and the other involving cosine. There's also an identity from advanced trigonometry that we want to use; a derivation is given in the notes if you want to read about it later. It says, to take the sine of a sum of angles Alpha and Beta, you multiply sine of Alpha by cos of Beta and add cos of Alpha times sine of Beta. We now sketch a proof that the derivative of sine x is cos x. Start off with a definition of the derivative, and this becomes the limit as h tends to zero of sine x plus h minus sine x, all over h. At this point, we rewrite the numerator using the advanced trigonometry identity we mentioned a moment ago, with angles x and h taking the place of Alpha and Beta. We now strategically reorganize the components of the numerator, and then use the fact that the limit of a sum is the sum of the limits to split this into two pieces. In the numerator of the first limit, we take out a common factor of sine x, and in the second limit, we have the factor cos x which behaves like a constant. So, it can come out the front. But now in the first piece, sine x is behaving like a constant, so it can come out the front. The first piece is just sine x times zero, and the second piece is just cos x times one, using the two limits from before and the whole thing simplifies to cos x. Which finally proves our claim. What's the derivative of cos x? The answer turns out to be negative sine x. You can convince yourself by looking at graphs, thinking about tangents to the cosine curve, and checking against corresponding points on the negative sine curve. The rigorous derivation using the limit definition appears in the notes. You're probably curious now about the derivative of tan x, you're not likely to guess the answer. It is in fact, the reciprocal of the square of cos x. You'll have to wait for a future video to see why. We showed before, that the derivative of e to the x is e to the x. We've just shown that the derivative of sine x is cos x, and mentioned that the derivative of cos x is minus sine x. This means that the circular functions are indestructible in the following sense, if f of x is sine x and the derivative is cos x, so each derivative, the second derivative, is negative sine x. So, each derivative, the third derivative, is negative cos x. So, its derivative, the fourth derivative, quickly become sine x, and we're back to where we started. Differentiating, add infinitum, just reproduces the same cycle of functions over and over again forever. Recall in an earlier video, we mentioned simple harmonic motion as an example of a displacement function, which happens to be the sine curve. Its derivative, the velocity is the cosine curve, its second derivative, the acceleration, is the negative sine curve, its third derivative, the jerk, is a negative cosine curve, it's fourth derivative, the snap, is the original sine curve, and the process goes on forever reproducing these four curves by differentiation. Today, we've really come a long way using the limit definition to rapidly expand our repertoire. We discussed and illustrated simple rules for differentiating polynomials, and noted, that by differentiating often enough we can make a polynomial disappear in the sense of going to zero. We explained why the natural exponential function reproduces a perfect copy of itself under differentiation. So, is in a certain sense indestructible. Sketched a proof that the derivative of sine is cosine, and mentioned that the derivative of cosine is negative sine, leading to an indestructible sequence of functions, sine, cosine, negative sine, and negative cosine, that reproduces itself forever under differentiation. 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.
