We derive some of the rules for computing derivatives.

1. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
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Notes
Sec on 2.3
Basic Differenta on Rules
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
. NYUMathematics
.
Notes
Announcements
Quiz 1 this week on
1.1–1.4
Quiz 2 March 3/4 on 1.5,
1.6, 2.1, 2.2, 2.3
Midterm Monday March
7 in class
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Notes
Objectives
Understand and use
these differen a on
rules:
the deriva ve of a
constant func on (zero);
the Constant Mul ple
Rule;
the Sum Rule;
the Difference Rule;
the deriva ves of sine
and cosine.
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2. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Recall: the derivative
Defini on
Let f be a func on 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 func on is said to be differen able at a and f′ (a) is the
deriva ve of f at a.
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Notes
The deriva ve …
…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 approxima on to f near a.
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Notes
Notation
Newtonian nota on Leibnizian nota on
dy d df
f′ (x) y′ (x) y′ f(x)
dx dx dx
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. 2
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3. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Link between the notations
f(x + ∆x) − f(x) ∆y dy
f′ (x) = lim = lim =
∆x→0 ∆x ∆x→0 ∆x dx
dy
Leibniz thought of as a quo ent of “infinitesimals”
dx
dy
We think of as represen ng a limit of (finite) difference
dx
quo ents, not as an actual frac on itself.
The nota on suggests things which are true even though they
don’t follow from the nota on per se
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Notes
Outline
Deriva ves so far
Deriva ves of power func ons by hand
The Power Rule
Deriva ves of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Mul ple Rule
Deriva ves of sine and cosine
.
.
Notes
Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (x).
Solu on
f(x + h) − f(x) (x + h)2 − x2
f′ (x) = lim = lim
h→0 h h→0 h
+ 2xh + h −
x2
2
x2
2x + h2
h
¡
= lim = lim
h→0 h h→0 h
= lim (2x + h) = 2x.
h→0
.
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. 3
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4. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
The second derivative
If f is a func on, so is f′ , and we can seek its deriva ve.
f′′ = (f′ )′
It measures the rate of change of the rate of change! Leibnizian
nota on:
d2 y d2 d2 f
2 2
f(x)
dx dx dx2
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Notes
The squaring function and its derivatives
y
f′
f′′ f increasing =⇒ f′ ≥ 0
f decreasing =⇒ f′ ≤ 0
. f x horizontal tangent at 0
=⇒ f′ (0) = 0
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Notes
Derivative of the cubing function
Example
Suppose f(x) = x3 . Use the defini on of deriva ve to find f′ (x).
Solu on
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5. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
The cubing function and its derivatives
No ce that f is increasing,
y and f′ 0 except
f′′ f′ f′ (0) = 0
No ce also that the
f tangent line to the graph
. x of f at (0, 0) crosses the
graph (contrary to a
popular “defini on” of
the tangent line)
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.
Notes
Derivative of the square root
Example
√
Suppose f(x) = x = x1/2 . Fnd f′ (x) with the defini on.
Solu on
√ √
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
(x + h) − x
¡ ¡ h
1
= lim (√ √ ) = lim (√ √ )= √
h→0 h x+h+ x h x+h+ x
h→0 2 x
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Notes
The square root and its derivatives
y
f Here lim+ f′ (x) = ∞ and f
′ x→0
f is not differen able at 0
. x
No ce also lim f′ (x) = 0
x→∞
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. 5
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6. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Derivative of the cube root
Example
√
Suppose f(x) = 3
x = x1/3 . Find f′ (x).
Solu on
.
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Notes
The cube root and its derivative
y
Here lim f′ (x) = ∞ and f
f x→0
is not differen able at 0
f′
. x No ce also
lim f′ (x) = 0
x→±∞
.
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Notes
One more
Example
Suppose f(x) = x2/3 . Use the defini on of deriva ve to find f′ (x).
Solu on
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. 6
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7. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
x → x2/3 and its derivative
y
f f is not differen able at 0
and lim± f′ (x) = ±∞
f′ x→0
. x No ce also
lim f′ (x) = 0
x→±∞
.
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Notes
Recap: The Tower of Power
y y′
x2 2x1 The power goes down by
x 3
3x2 one in each deriva ve
1 −1/2 The coefficient in the
x1/2 2x deriva ve is the power of
1 −2/3
x1/3 3x
the original func on
2 −1/3
x2/3 3x
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Notes
The Power Rule
There is moun ng evidence for
Theorem (The Power Rule)
Let r be a real number and f(x) = xr . Then
f′ (x) = rxr−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.
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. 7
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8. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Outline
Deriva ves so far
Deriva ves of power func ons by hand
The Power Rule
Deriva ves of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Mul ple Rule
Deriva ves of sine and cosine
.
.
Notes
Remember your algebra
Fact
Let n be a posi ve whole number. Then
(x + h)n = xn + nxn−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 xk hn−k
n copies k=0
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Notes
Pascal’s Triangle
.
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 = 1x2 + 2xh + 1h2
1 5 10 10 5 1
(x + h)3 = 1x3 + 3x2 h + 3xh2 + 1h3
1 6 15 20 15 6 1 ... ...
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9. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Proving the Power Rule
Theorem (The Power Rule)
d n
Let n be a posi ve whole number. Then x = nxn−1 .
dx
Proof.
As we showed above,
(x + h)n = xn + nxn−1 h + (stuff with at least two hs in it)
(x + h)n − xn nxn−1 h + (stuff with at least two hs in it)
So =
h h
= nxn−1 + (stuff with at least one h in it)
and this tends to nxn−1 as h → 0.
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Notes
The Power Rule for constants?
d 0
Theorem like x = 0x−1
d dx
Let c be a constant. Then c = 0.
.
dx
Proof.
Let f(x) = c. Then
f(x + h) − f(x) c − c
= =0
h h
So f′ (x) = lim 0 = 0.
h→0
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Notes
Calculus
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. 9
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10. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Recall the Limit Laws
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. . . .
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Notes
Adding functions
Theorem (The Sum Rule)
Let f and g be func ons and define
(f + g)(x) = f(x) + g(x)
Then if f and g are differen able at x, then so is f + g and
(f + g)′ (x) = f′ (x) + g′ (x).
Succinctly, (f + g)′ = f′ + g′ .
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Notes
Proof of the Sum Rule
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)
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. 10
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11. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Scaling functions
Theorem (The Constant Mul ple Rule)
Let f be a func on and c a constant. Define
(cf)(x) = cf(x)
Then if f is differen able at x, so is cf and
(cf)′ (x) = c · f′ (x)
Succinctly, (cf)′ = cf′ .
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Notes
Proof of Constant Multiple Rule
Proof.
Again, follow your nose.
(cf)(x + h) − (cf)(x) cf(x + h) − cf(x)
(cf)′ (x) = lim = lim
h→0 h h→0 h
f(x + h) − f(x) ′
= c lim = c · f (x)
h→0 h
.
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Notes
Derivatives of polynomials
Example
d ( 3 )
Find 2x + x4 − 17x12 + 37
dx
Solu on
d ( 3 ) d ( 3) d d ( ) d
2x + x4 − 17x12 + 37 = 2x + x4 + −17x12 + (37)
dx dx dx dx dx
d d d
= 2 x3 + x4 − 17 x12 + 0
dx dx dx
= 2 · 3x2 + 4x3 − 17 · 12x11
= 6x2 + 4x3 − 204x11
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. 11
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12. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Notes
Outline
Deriva ves so far
Deriva ves of power func ons by hand
The Power Rule
Deriva ves of polynomials
The Power Rule for whole number powers
The Power Rule for constants
The Sum Rule
The Constant Mul ple Rule
Deriva ves of sine and cosine
.
.
Notes
Derivatives of Sine and Cosine
Fact
d
sin x = cos x
dx
Proof.
From the defini on:
d sin(x + h) − sin x ( sin x cos h + cos x sin h) − sin x
sin x = lim = lim
dx h→0 h 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
.
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Angle addition formulas Notes
See Appendix A
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13. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Two important trigonometric Notes
limits
See Section 1.4
sin θ
lim . =1
θ→0 θ
cos θ − 1
sin θ θ lim =0
θ θ→0 θ
.
−1 1 − cos θ 1
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Notes
Illustration of Sine and Cosine
y
. x
π −π 0 π π cos x
2 2
sin x
f(x) = sin x has horizontal tangents where f′ = cos(x) is zero.
what happens at the horizontal tangents of cos?
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Notes
Derivative of Cosine
Fact
d
cos x = − sin x
dx
Proof.
We already did the first. The second is similar (muta s mutandis):
d cos(x + h) − cos x (cos x cos h − sin x sin h) − cos x
cos x = lim = lim
dx h→0 h 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
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. 13
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14. . V63.0121.001: Calculus I
. Sec on 2.3: Basic Differenta on Rules
.
Summary Notes
What have we learned today?
The Power Rule
The deriva ve of a sum is the sum of the deriva ves
The deriva ve of a constant mul ple of a func on is that
constant mul ple of the deriva ve
The deriva ve of sine is cosine
The deriva ve of cosine is the opposite of sine.
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Notes
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Notes
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