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,
dx [c] = 0
when c is just any ol’ number.
Example:  = 0,
dx 2 = 0,
[▯] = 0.
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.
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
where m is the combinatorial function n choose m, de▯ned as
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
You don’t need to know this, but I had to put it here, so that we can go on with showing what the
derivative of x is! So, let’s begin:
0 f(x + h) ▯ f(x)
f (x) = lim
(x + h) ▯ x
x + nxn▯1h + n(n▯1)xn▯2h + ▯▯▯ + nxh▯1 + h ▯ x n
= lim 2!
nxn▯1h + 2! xn▯2h + ▯▯▯ + nxhn▯1+ hn
h!0 h h i
n▯1 n(n▯1)n▯2 n▯2 n▯1
h nx + 2! x h + ▯▯▯ + nxh + h
n▯1 n(n ▯ 1) n▯2 n▯2 n▯1
= lim nx + x h + ▯▯▯ + nxh + h
= nx n▯1
Oh, wow! So x = nx n▯1!!! Just take the exponent down, multiply it in front, and then
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
Example: x = 3x 2
x▯2 = ▯2x ▯3
d x = ▯x ▯▯1
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)]
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
d ▯ 2
Example: dx 3x = 3 ▯ 2x = 6x