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WEEK 3
Differentiation I
3.1
Year 110
Motivations : Tangents and velocities
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N. Sato & K.-W. Tsoi
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Definition 3.1.1 (Derivative). The derivative f Õ (a) of f (x) at x = a is defined to be
(if exists) the slope of tangent at x = a. To be specific,
f Õ (a) := lim
hæ0
f (a + h) ≠ f (a)
.
h
In this case, we say that f (x) is differentiable at x = a.
Example (Velocity). The displacement of a particle can be described by the function
f (t) = sin(sin(sin(t))).
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htuatx-o.tn
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What is the initial velocity of the particle?
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Calculus 1 - Year 110
3.2
N. Sato & K.-W. Tsoi
Examples of derivatives
Example. Compute, by definition, the derivative f Õ (x) of f (x) = sin(x).
Recall
①
所
1-asxnop.tt
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篼
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hinsinxlcahtl-cosxsinhh.no
h-lim-sinx.1-ahh-ax.hn
hi-sinxo-cax.li
mo
cosxxdqio.fi
。
fM-sinxfkx.cat
34
Calculus 1 - Year 110
N. Sato & K.-W. Tsoi
Example (Exotic sine curve2 ). Consider the function defined by
Y
]x2 sin 1
x
f (x) =
[0
(a) Is f (x) continuous at x = 0 ?
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if x ”= 0
if x = 0
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(b) Is f (x) differentiable at x = 0 ? If so, compute f Õ (0).
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35
Calculus 1 - Year 110
wnd
3.3
36
N. Sato & K.-W. Tsoi
tangent 徘
Examples of non-differentiable functions
洲
Example (Vertical slope). Prove that f (x) = x1/5 is not differentiable at x = 0.
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Figure 3. The graph y = x1/5
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Example (Absolute value). Prove that f (x) = |x| is not differentiable at x = 0.
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Calculus 1 - Year 110
N. Sato & K.-W. Tsoi
Example (Exotic sine curve1 ). Consider the function defined by
Y
]x sin 1
x
f (x) =
[0
if x ”= 0
if x = 0
.
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Is f (x) differentiable at x = 0 ?
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BNOTdhiableatx.co/Rmark.CanparetiswMhExoticsnai/
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,
37
Calculus 1 - Year 110
3.4
N. Sato & K.-W. Tsoi
Properties of differentiable functions
Theorem 3.4.1 (Important limit). lim
hæ0
3
eh ≠ 1
h
4
= 1.
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Theorem 3.4.2 (Differentiability implies continuity).
If a function f (x) is differentiable at x = a, then f (x) is continuous at x = a.
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38
𤄿
Calculus 1 - Year 110
3.5
N. Sato & K.-W. Tsoi
Standard derivatives : powers, trigonometry, ex
Table for standard derivatives (Part I)
f (x)
Powers
f Õ (x)
c (constants)
f Õ (x)
0
kx
xk
Trigonometry
f (x)
kianumber.ca
""
sin(x)
x
csc(x)
cos(x)
sinx
sec(x)
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tan(x)
cot(x)
Ex
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Exponential
-
ex
Theorem 3.5.1 (Integral+ power rule). Let n be a positive integer and f (x) = xn .
Then f Õ (x) = nxn≠1 .
Ülim 挑以利
htuhinlxthnx lnoh. n#tGT!htYjunH-xnhn
h
嗎 nxntthljunkltn-n.it※
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Theorem 3.5.2 (Exponential). Let f (x) = ex . Then f Õ (x) = ex .
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39
Calculus 1 - Year 110
3.6
N. Sato & K.-W. Tsoi
Sum, product and quotient rules
Theorem 3.6.1 (Algebra of derivatives). Given f (x) and g(x) to be two differentiable
functions. Then the following are true.
(a) (f (x) ± g(x))Õ = f Õ (x) ± g Õ (x),
(b) (f (x)g(x))Õ = f Õ (x)g(x) + f (x)g Õ (x), hutmle
(c) If g(x) ”= 0, then
3
f (x)
g(x)
4Õ
=
f Õ (x)g(x) ≠ f (x)g Õ (x)
2
(g(x))
2
Example. Let y = xe · ex . Compute
.
Quditruk
dy
.
dx
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Theorem 3.6.2 (Tangent). Let f (x) = tan(x). Then f Õ (x) = sec2 (x).
fp.in/asxf4y=C0sxasx- ixtsnyc s2X=l+tan2x-sec2x.
,
40
Calculus 1 - Year 110
N. Sato & K.-W. Tsoi
Example. Consider the function f (x) = x4 ex + 3x + 2.
(a) Compute f Õ (x).
(b) Find the equation of tangent at x = 0.
(c) Find the equation of normal at x = 0.
(a)
拟 妒城 侧州
(C)
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。
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Hence
,
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normal
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列
41
Calculus 1 - Year 110
N. Sato & K.-W. Tsoi
Proof. [Proof of Product Rule]
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hofLHS-lfgi-fhglxthl-fM.gl#h=limfMH9NtH-flD9lt h14fMgath1-f1
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42
Calculus 1 - Year 110
3.7
N. Sato & K.-W. Tsoi
43
More examples
Example. Consider the following function f (x).
f (x) =
Y
_
_
]cos(x)
if x < 0
。
.
_
_
[a sin(x) + b
if x Ø 0
It is known that f (x) is differentiable at x = 0. What are the values of a and b?
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Calculus 1 - Year 110
44
N. Sato & K.-W. Tsoi
Differentilheexistenceofhmfuhtflhl
abhgoffhatx-aconc.ms
Example (Common misconception). Consider the following function.
f (x) =
Y
2
_
_x ≠ 4x
]
h→ 0
if x Ø 2
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.
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_
_
[x + x
e2
ex
if x < 2
Prove that lim+ f Õ (x) = lim f Õ (x) = 0. Despite this, explain why f (x) is not
xæ2≠
xæ2
differentiable at x = 2.
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Calculus 1 - Year 110
N. Sato & K.-W. Tsoi
Example. Let f (x) be a function with the following properties.
(1) f (x + y) = ex f (y) + ey f (x) for all real numbers x, y.
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f (h)
(2) lim
= 2021.
hæ0 h
isago.nfcy-eifite0.fi
ntinumsldf a nbkatal . a/Putx-y-ointopnpertgcn
Answer the following questions.
(a) Find f (0) and compute lim f (h). Is f continuous at x = 0?
hæ0
(b) Prove that f is differentiable everywhere and that f Õ (x) = 2021ex + f (x).
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45