In earlier videos, we've introduced and developed the limit definition of the derivative formally expressed in the notation of functions, so that if f is a function, then the derivative of f of x with respect to x is denoted by f dashed of x, with a dash as a superscript. In today's video, we introduce and apply a very commonly used notation for the derivative called dy dx, that looks like a fraction and is known as Leibniz's notation. In fact, dy dx is still formally speaking a limit, but the idea of thinking of the derivative as a fraction and deliberately using notation as a fraction goes back to the pioneering work of Gottfried Leibniz, one of the founders in parallel to Isaac Newton of calculus in the 17th century, and after whom this notation is named. Leibniz thought in terms of very powerful underlying heuristics. He considered the derivative to be a fraction involving infinitesimals, which idealized infinitely small numbers that interact using an arithmetic that extends that of the real number system. His notation was so carefully chosen that it has survived intact for hundreds of years. It's interesting that Leibniz's number system wasn't placed on a formal rigorous footing until the 1960s, which is not so long ago. If you want to read more about this fascinating topic, you can look up terms on the Internet such as nonstandard analysis or the hyperreal number system. We begin by revisiting a familiar diagram intended to represent a general curve, y equals f of x, and a point of interest with coordinates x f of x, and a tangent line to the curve at that point, and a nearby point with coordinates x plus h, f of x plus h. We connect the two points with a straight line segment called a secant, the horizontal runners h and the vertical rise is f of x plus h minus f of x. We think of adding h to x as a slight variation or perturbation of input, and we introduce some new notation and give h the name Delta x. Delta being the Greek capital letter corresponding to the Latin d, and think of Delta for difference. So, Delta x equal to h is the difference in the x-coordinate, and we may simply say difference in x or change in x. As we perturb the x-coordinate, this induces a perturbation in the y-coordinate, and the change in effect in the vertical direction f of x plus h minus f of x, we may now rename Delta y, and simply say difference in y or change in y. The slope of the secant that we recognized before as f of x plus h minus f of x of h, now becomes, using this notation, Delta y over Delta x. The change in y over the change in x. The slope of the tangent line that we recognized as limiting slope of the secant can now be expressed using this new notation as the limit as Delta x goes to zero of Delta y divided by Delta x. We can now rewrite the limit definition of the derivative of y equals f of x, or equivalently as the limit as Delta x goes to zero of Delta y divided by Delta x. This has a very beautiful notational abbreviation dy over dx, just spoken dy dx and called Leibniz notation in honor of Gottfried Leibniz. Informally, in the limit, you can think of the great Delta x turning into the Latin lowercase dx, and the Greek Delta y turning into the Latin lowercase dy. Leibniz thought of dx and dy as idealized mathematical objects, representing some kind of infinitely small numbers that had their own arithmetic, that paralleled the arithmetic of the real numbers. In modern terminology, dx and dy are called differentials. They become very useful heuristic devices,both in applications and also later in the final module when we manipulate integrals. Let's revisit some earlier examples, but interpreted or expressed using Leibniz notation. First, if y equals k is a constant function, then the graph is a horizontal line with no change in the y-value, giving slope zero everywhere. So, dy dx is zero. Next, if y equals mx plus k is a linear function, where m is the slope and k is y-intercept, then y changes m units uniformly for each unit that exchanges. So, dy dx is the constant m. We know that previously that the derivative is additive, which means that the derivative of a sum is the sum of the derivatives. For example, if y is a quadratic function at x, say x squared plus bx plus c, where a, b, and c are constants, then dy dx is the sum of the derivatives of each of the pieces, which can be expressed in this way. So, the derivative of ax squared, regarded as root for the function that takes x to x squared, can be rewritten as d dx of ax squared, which is 2ax. The derivative of bx can be written as d dx of bx, which is b, and the derivative of the constant function c can be rewritten as d dx of c, which is zero. The entire derivative simplifies to 2ax plus b. We mentioned in general that the derivative of x dn is nx to the n minus one, for any fixed exponent n. We carefully proved that the derivative of e to the x is e to the x, reproducing itself exactly under differentiation. Notice that the variable x is the exponent. which is a common error by students to confuse the exponential rule for differentiation with the previous rule and guess that the answer for the derivative of e to the x should be the result of bringing the exponent x down to the front, and then creating a new exponent by subtracting one, which is completely wrong and off track. This is a subtle error and the result of confusing the contrasting roles of the variable x used in building power functions, where x is a base and exponential functions, where x is an exponent. We've sketched a proof that the derivative of sine x is cos x, and mentioned that the derivative of cos x is minus sine x. Good notation can be amazingly transformative, You might recall we developed the quadratic formula really from scratch. Another derivation at the time to quite a few lines and some tricks, it was relatively painless using algebraic notation. Imagine as a medieval mathematician, without our notation, trying to express everything in mixtures of words and hybrids of symbols and trying to avoid any heresie associated with using negative numbers. Leibniz notation, dy dx, is truly a miracle of inventiveness. So, simple, yet so powerful. It places emphasis on the roles of the variables x and y, where the differential associated with x, appears in the denominator of some kind of fraction and the differential dy, in the numerator. These distinctions are invisible or opaque, in the function notation, f dashed x, for the derivative. If one wants to reverse the roles of the variables x and y, it makes perfectly, good sense using Leibniz notation, to tip dy dx upside down, that is form dx dy, interchanging the differentials in the numerator and the denominator. If these were ordinary fractions, the effect would be to form the reciprocal. This equation dx dy equals the reciprocal of dy dx, is an irresistible deduction from the notation and intuition of Leibniz and turns out to be a theorem about derivatives of inverse functions. Interchanging the roles of x and y of input and output, has the effect of inverting the original function. So, this formula tells us how to differentiate the inverse function. It becomes a nontrivial and useful theorem about derivatives. Why is it useful? Let's use this idea to find the derivative of the natural logarithm function, knowing that it's the inverse of the natural exponential function. Start off with y equals e to the x for which we already know the derivative, which is e to the x. We want information about the inverse function. So, tip the derivative upside down, which both reverses the roles of x and y and also reciprocates the derivative dy dx, of the original exponential function. This is just one over e to the x, which is one over y. But y is e to the x so undoing this gives x equals the natural logarithm of y. So, saying that dx dy equals one over y, is the same as saying, d dy of log of y, is one over y. We now revert back using x as the symbol for the input. We get that the derivative of log of x with respect to x, is the reciprocal of x. Amazing, such a simple and elegant answer! The formula makes sense visually. Here are the familiar natural logarithm and exponential functions on the same diagram and we get from one to the other by reflecting in the line y equals x. Euler's number e was chosen so that the slope of the tangent to the curve, y equals e to the x at the y intercept, is one. So, this tangent is parallel to the line y equals x. When you do the reflection, the reflected tangent line, now touching the curve y equals log x at the x intercept, retains the same slope. Both of these tangent lines, have slope one. Of course one is one over one. So, this matches the formula for the derivative of log of x. You can visualize the tangent to the curve at x equals two and if your diagram is accurate, you will see, that it has a slope of a half, also matching the formula for the derivative. We can do the same thing for x equals three and get a slope of a third and anywhere on the x axis you'll find the slope of the tangent to the curve is the reciprocal of x. Let's try this idea, tipping the Leibniz derivative upside down, to differentiate the square root function. Start off with the squaring function, y equals x squared and focus on x, greater than or equal to 0 so, that x is just the square root of y. We know the derivative of y with respect to x is just 2 x and this is two times the square root of y. We now form dx dy, which is the reciprocal of the dy dx, which becomes one over two times the square root of y. But dx dy is just d dy of the square root of y, so we can express everything in terms of y. We now revert to using the symbol x as a typical input. So, that d dx of the square root of x, is one over two times the square root of x. So, we discovered that, to differentiate the square root of x, you just double it and reciprocate. Not something you're ever likely to guess. We can relate this to fractional powers because the square root of x is just x to the power of one half. This rule for differentiation then becomes dy dx of x to the power of one half, is one over two times x to the power of one half. Note that this can be rewritten, as one half of x to the power of minus one half and so, we get this nice formula. Note that, this match is precisely the formula that we've mentioned before, for the derivative of x to the n, where n equals one half. Let's finish by differentiating the cube root function using this inversion trick, but y equals x cubed. So, x becomes the cube root of y, but we proved a while ago that, dy dx is 3 x squared. Which becomes three times the square of y to the third, which is three times y, to the two thirds. So, flipping the derivative over, we get dx dy is one over three times y to the two thirds. Which can be rewritten as one third of y, to the minus two-thirds. At the same time, dx dy is just d dy of y to the one third. Reverting to x as a typical input, we get d dx of x to the one third, is one third of x to the minus two thirds. Happily, this is consistent with the formula for the derivative of x to the n, when n equals one third. So, everything is working nicely. In today's video, we introduced and practiced the Leibniz notation for the derivative called dy dx, which appears as a fraction involving the differentials dx and dy. Even though the derivatives formally defined as a limit of fractions, thinking of it as behaving like a fraction in its own right, turns out to be very useful especially for applications and for making heuristic leaps of the imagination. Once such leap, is to invert dy dx to get dx dy which then can be used to get information about the derivative of the associated inverse function. We exploited this trick, to discover that the derivative of the natural logarithm function, is the reciprocal function and also to differentiate the square root and cube root functions. 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.
