Lesson 8: Basic Differentiation Rules (Section 41 handout)

V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010

Notes
Section 2.3
Basic Differentiation Rules

V63.0121.041, Calculus I

New York University

September 28, 2010

Announcements

Last chance for extra credit on Quiz 1: Do the get-to-know you
survey and photo by October 1.

Announcements
Notes

Last chance for extra credit
on Quiz 1: Do the
get-to-know you survey and
photo by October 1.

V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 2 / 42

Objectives
Notes

Understand and use these
differentiation rules:
the derivative of a
constant function (zero);
the Constant Multiple
Rule;
the Sum Rule;
the Difference Rule;
the derivatives of sine and
cosine.


1


Recall: the derivative
Notes

Definition
Let f be a function and a a point in the domain of f . If the limit

f (a + h) − f (a) f (x) − f (a)
f (a) = lim = lim
h→0 h x→a x −a

exists, the function is said to be differentiable at a and f (a) is the
derivative of f at a.
The derivative . . .
. . . measures the slope of the line through (a, f (a)) tangent to the
curve y = f (x);
. . . represents the instantaneous rate of change of f at a
. . . produces the best possible linear approximation to f near a.


Notation
Notes

Newtonian notation Leibnizian notation
dy d df
f (x) y (x) y f (x)
dx dx dx


Link between the notations
Notes

f (x + ∆x) − f (x) ∆y dy
f (x) = lim = lim =
∆x→0 ∆x ∆x→0 ∆x dx

dy
Leibniz thought of as a quotient of “infinitesimals”
dx
dy
We think of as representing a limit of (finite) difference quotients,
dx
not as an actual fraction itself.
The notation suggests things which are true even though they don’t
follow from the notation per se


2


Outline
Notes

Derivatives so far
Derivatives of power functions by hand
The Power Rule

Derivatives of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Multiple Rule

Derivatives of sine and cosine


Derivative of the squaring function
Notes

Example
Suppose f (x) = x 2 . Use the deﬁnition of derivative to ﬁnd f (x).

Solution

f (x + h) − f (x) (x + h)2 − x 2
f (x) = lim = lim
h→0 h h→0 h
x2 2 x2
+ 2xh + h −
2
2x h + h¡
¡
= lim = lim
h→0 h h→0 h
¡
= lim (2x + h) = 2x.
h→0

So f (x) = 2x.


The second derivative
Notes

If f is a function, so is f , and we can seek its derivative.

f = (f )

It measures the rate of change of the rate of change! Leibnizian notation:

d 2y d2 d 2f
f (x)
dx 2 dx 2 dx 2


3


The squaring function and its derivatives
Notes

y
f
f f increasing =⇒ f ≥ 0
f f decreasing =⇒ f ≤ 0
x horizontal tangent at 0
=⇒ f (0) = 0


Derivative of the cubing function
Notes
Example
Suppose f (x) = x 3 . Use the definition of derivative to find f (x).

Solution


The cubing function and its derivatives
Notes

y Notice that f is increasing,
f f and f > 0 except f (0) = 0
Notice also that the tangent
f line to the graph of f at
x (0, 0) crosses the graph
(contrary to a popular
“definition” of the tangent
line)


4


Derivative of the square root function
Notes
Example
√
Suppose f (x) = x = x 1/2 . Use the definition of derivative to find f (x).

Solution

√ √
f (x + h) − f (x) x +h− x
f (x) = lim = lim
h→0 h h→0 h
√ √ √ √
x +h− x x +h+ x
= lim ·√ √
h→0 h x +h+ x
(& + h) − &
x x h
√ √
¡
= lim √ = lim √
h→0 h x +h+ x h→0 h
¡ x +h+ x
1
= √
2 x
√
So f (x) = x = 1 x −1/2 .
2


The square root function and its derivatives
Notes

y

f Here lim+ f (x) = ∞ and f
x→0
f is not differentiable at 0
x
Notice also lim f (x) = 0
x→∞


Derivative of the cube root function
Notes
Example
√
Suppose f (x) = 3
x = x 1/3 . Use the definition of derivative to find f (x).

Solution


5


The cube root function and its derivatives
Notes

y

Here lim f (x) = ∞ and f is
f x→0
not differentiable at 0
f
x Notice also lim f (x) = 0
x→±∞


One more
Notes

Example
Suppose f (x) = x 2/3 . Use the definition of derivative to find f (x).

Solution


The function x → x 2/3 and its derivative
Notes

y

f is not differentiable at 0
f and lim f (x) = ±∞
x→0±
f
x Notice also lim f (x) = 0
x→±∞


6


Recap: The Tower of Power
Notes

y y
x2 2x 1 The power goes down by
x3 3x 2 one in each derivative
1 −1/2 The coefficient in the
x 1/2 2x derivative is the power of
1 −2/3
x 1/3 3x the original function
2 −1/3
x 2/3 3x


The Power Rule
Notes

There is mounting evidence for
Theorem (The Power Rule)
Let r be a real number and f (x) = x r . Then

f (x) = rx r −1

as long as the expression on the right-hand side is defined.

Perhaps the most famous rule in calculus
We will assume it as of today
We will prove it many ways for many different r .


The other Tower of Power
Notes


7


Outline
Notes

Derivatives so far
The Power Rule

The Sum Rule



Remember your algebra
Notes
Fact
Let n be a positive whole number. Then

(x + h)n = x n + nx n−1 h + (stuff with at least two hs in it)

Proof.
We have
n
(x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = ck x k hn−k
n copies k=0

n
The coefficient of x is 1 because we have to choose x from each
binomial, and there’s only one way to do that. The coefficient of x n−1 h is
the number of ways we can choose x n − 1 times, which is the same as the
number of different hs we can pick, which is n.

Pascal’s Triangle
Notes

1

1 1

1 2 1

1 3 3 1
(x + h)0 = 1
1 4 6 4 1 (x + h)1 = 1x + 1h
(x + h)2 = 1x 2 + 2xh + 1h2
1 5 10 10 5 1
(x + h)3 = 1x 3 + 3x 2 h + 3xh2 + 1h3
... ...
1 6 15 20 15 6 1


8


Proving the Power Rule
Notes
Theorem (The Power Rule)
Let r be a positive whole number. Then
d r
x = rx r −1
dx

Proof.
As we showed above,

(x + h)n = x n + nx n−1 h + (stuff with at least two hs in it)

So
(x + h)n − x n nx n−1 h + (stuff with at least two hs in it)
=
h h
= nx n−1 + (stuff with at least one h in it)

and this tends to nx n−1 as h → 0.

Notes

Theorem d 0
like x = 0x −1
Let c be a constant. Then dx
d
c=0
dx

(although x → 0x −1 is not defined at zero.)
Proof.
Let f (x) = c. Then

f (x + h) − f (x) c −c
= =0
h h
So f (x) = lim 0 = 0.
h→0


Calculus
Notes

9


Recall the Limit Laws
Notes

Fact
Suppose lim f (x) = L and lim g (x) = M and c is a constant. Then
x→a x→a
1. lim [f (x) + g (x)] = L + M
x→a
2. lim [f (x) − g (x)] = L − M
x→a
3. lim [cf (x)] = cL
x→a
4. . . .


Adding functions
Notes

Theorem (The Sum Rule)
Let f and g be functions and define

(f + g )(x) = f (x) + g (x)

Then if f and g are differentiable at x, then so is f + g and

(f + g ) (x) = f (x) + g (x).

Succinctly, (f + g ) = f + g .


Proof of the Sum Rule
Notes

Proof.
Follow your nose:

(f + g )(x + h) − (f + g )(x)
(f + g ) (x) = lim
h→0 h
f (x + h) + g (x + h) − [f (x) + g (x)]
= lim
h→0 h
f (x + h) − f (x) g (x + h) − g (x)
= lim + lim
h→0 h h→0 h
= f (x) + g (x)

Note the use of the Sum Rule for limits. Since the limits of the difference
quotients for for f and g exist, the limit of the sum is the sum of the limits.


10


Scaling functions
Notes

Theorem (The Constant Multiple Rule)
Let f be a function and c a constant. Deﬁne

(cf )(x) = cf (x)

Then if f is diﬀerentiable at x, so is cf and

(cf ) (x) = c · f (x)

Succinctly, (cf ) = cf .


Proof of the Constant Multiple Rule
Notes

Proof.
Again, follow your nose.

(cf )(x + h) − (cf )(x)
(cf ) (x) = lim
h→0 h
cf (x + h) − cf (x)
= lim
h→0 h
f (x + h) − f (x)
= c lim
h→0 h
= c · f (x)


Notes
Example
d
Find 2x 3 + x 4 − 17x 12 + 37
dx

Solution

d
2x 3 + x 4 − 17x 12 + 37
dx
d d 4 d d
= 2x 3 + x + −17x 12 + (37)
dx dx dx dx
d d 4 d
= 2 x3 + x − 17 x 12 + 0
dx dx dx
= 2 · 3x 2 + 4x 3 − 17 · 12x 11
= 6x 2 + 4x 3 − 204x 11


11


Outline
Notes

Derivatives so far
The Power Rule

The Sum Rule



Derivatives of Sine and Cosine
Notes
Fact

d
sin x = cos x
dx

Proof.
From the deﬁnition:
d sin(x + h) − sin x
sin x = lim
dx h→0 h
( sin x cos h + cos x sin h) − sin x
= lim
h→0 h
cos h − 1 sin h
= sin x · lim + cos x · lim
h→0 h h→0 h
= sin x · 0 + cos x · 1 = cos x


Angle addition formulas
See Appendix A Notes


12


Two important trigonometric limits
See Section 1.4 Notes

sin θ
lim =1
θ→0 θ
sin θ θ cos θ − 1
lim =0
θ θ→0 θ

1 − cos θ 1


Illustration of Sine and Cosine
Notes

y

x
π −π 0 π π
2 2 cos x
sin x

f (x) = sin x has horizontal tangents where f = cos(x) is zero.
what happens at the horizontal tangents of cos?


Derivatives of Sine and Cosine
Notes
Fact

d d
sin x = cos x cos x = − sin x
dx dx

Proof.
We already did the ﬁrst. The second is similar (mutatis mutandis):

d cos(x + h) − cos x
cos x = lim
dx h→0 h
(cos x cos h − sin x sin h) − cos x
= lim
h→0 h
cos h − 1 sin h
= cos x · lim − sin x · lim
h→0 h h→0 h
= cos x · 0 − sin x · 1 = − sin x


13


What have we learned today?
Notes

The Power Rule
The derivative of a sum is the sum of the derivatives
The derivative of a constant multiple of a function is that constant
multiple of the derivative
The derivative of sine is cosine
The derivative of cosine is the opposite of sine.


Notes

Notes

14

Lesson 8: Basic Differentiation Rules (Section 41 handout)

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Lesson 8: Basic Differentiation Rules (Section 41 handout)