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3.3 techniques of differentiation.pdf

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University of Toronto St. George
Dan Dolderman

Calc 1 Wed 9/24/08 3.3 - Techniques of Di▯erentiation Let us begin today’s lecture by recalling the derivative of a constant. Remember how if you have a line y = mx + b, then the derivative is just the slope of that line. This means that a a constant function f(x) = c for any number c has slope zero (it’s just a horizontal line!), so whenever you take the derivative of a constant, it’s zero. ie, d dx [c] = 0 when c is just any ol’ number. d Example: [2] = 0, dx ▯ ▯ d ▯3 dx 2 = 0, d [▯] = 0. dx n Sweet. Now what about a power function? ie, what’s the derivative of x for n an integer? Unfortunately, this proof uses a theorem you probably have no seen before, called the binomial theorem. So, here’s a small introduction to that. n Binomial Theorem: tells us how to expand out (x + y) for any n an integer. ▯ ▯ ▯ n ▯ n ▯ n n ▯n▯ (x + y) = x + xn▯1y + xn▯2y + ▯▯▯ + xy n▯1+ yn 0 1 2 n ▯ 1 n ▯n▯ where m is the combinatorial function n choose m, de▯ned as ▯ ▯ n n! m = m!(n ▯ m)! So, rewriting the above, we get n n n▯1 n(n ▯ 1) n▯2 2 n▯1 n (x + y) = x + nx y + x y + ▯▯▯ + nxy + y 2! You don’t need to know this, but I had to put it here, so that we can go on with showing what the n derivative of x is! So, let’s begin: 0 f(x + h) ▯ f(x) f (x) = lim h!0 h (x + h) ▯ x = lim h!0 h x + nxn▯1h + n(n▯1)xn▯2h + ▯▯▯ + nxh▯1 + h ▯ x n = lim 2! h!0 h n(n▯1) nxn▯1h + 2! xn▯2h + ▯▯▯ + nxhn▯1+ hn = lim h!0 h h i n▯1 n(n▯1)n▯2 n▯2 n▯1 h nx + 2! x h + ▯▯▯ + nxh + h = lim h!0 h n▯1 n(n ▯ 1) n▯2 n▯2 n▯1 = lim nx + x h + ▯▯▯ + nxh + h h!0 2! = nx n▯1 1 d Oh, wow! So x = nx n▯1!!! Just take the exponent down, multiply it in front, and then dx subtract the exponent by one! Wow! That’s going to save a lot of work and time! And guess what! This doesn’t have to be just for n an integer! This can work for any real number n! ...The proof of why it works for any real number n comes when we do section 4.1 (implicit di▯), so just believe me for now d Example: x = 3x 2 dx d x▯2 = ▯2x ▯3 dx d 3 x3=2= x1=2 dx 2 d x = ▯x ▯▯1 dx What other fun things can we do with derivatives? We can multiply the function by a constant! And this gives us d [cf(x)] = c [f(x)] dx dx In other words, if you multiply a function by some constant, some number, then you can just ignore it for the time being and take the derivative of the function itself, and then multiply by the constant afterwards. d ▯ 2 Example: dx 3x = 3 ▯ 2x = 6x p d
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