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![Page 1: Derivatives (1A) - Wikimedia · Derivatives (1A) 17 Young Won Lim 12/29/15 The differential of a function ƒ(x) of a single real variable x is the function of two independent real](https://reader033.vdocuments.us/reader033/viewer/2022050308/5f702b506d2c386e565d470f/html5/thumbnails/1.jpg)
Young Won Lim12/29/15
Derivatives (1A)
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Young Won Lim12/29/15
Copyright (c) 2011 - 2014 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
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Derivatives (1A) 3 Young Won Lim12/29/15
Differentials
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Derivatives (1A) 4 Young Won Lim12/29/15
A triangle and its slope
y = f x
f x1 h − f x1
h
x1, f x1
x1 h , f x1 h
x1, f x1
http://en.wikipedia.org/wiki/Derivative
x1 x1+h
f (x1+h)
f (x1)
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Derivatives (1A) 5 Young Won Lim12/29/15
Many smaller triangles and their slopes
f ( x1)
(x1, f (x1))
f ( x1+h2)
f ( x1+h1)
f ( x1+h)
f (x1 + h) − f ( x1)
h
f (x1 + h1) − f (x1)
h1
f (x1 + h2) − f (x1)
h2
limh→ 0
f (x1 + h) − f (x1)
h
x1 x1+h2 x1+h1 x1+h
(x1+h , f (x1+h))
h2
h1h
(x1, f (x1))
(x1+h , f (x1+h))
http://en.wikipedia.org/wiki/Derivative
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Derivatives (1A) 6 Young Won Lim12/29/15
The limit of triangles and their slopes
y = f x
f ' (x1) = limh→0
f (x1 + h) − f (x1)
h
The derivative of the function f at x1
http://en.wiktionary.org/
f ' (x) = limh→0
f (x + h) − f (x)h
=dfdx
=ddx
f (x)
The derivative function of the function f
y ' = f ' (x)
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Derivatives (1A) 7 Young Won Lim12/29/15
The derivative as a function
y = f x x1
x2
x3
f x1
f x2
f x3
f x
x1
x2
x3
f ' x 1
f ' x 2
f ' x 3
f ' x
= limh→0
f (x + h) − f (x)h
y ' = f ' (x)
Derivative Function
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Derivatives (1A) 8 Young Won Lim12/29/15
The notations of derivative functions
x1
x2
x3
f ' x 1
f ' x 2
f ' x 3
f ' (x)
limh→0
f (x + h) − f (x )
h
y ' = f ' x
Largrange's Notation
dydx
=ddx
f (x)
Leibniz's Notation
y = f x
Newton's Notation
D x y = D x f x
Euler's Notation
not a ratio.
● derivative with respect to x● x is an independent variable
slope of a tangent line
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Derivatives (1A) 9 Young Won Lim12/29/15
Another kind of triangles and their slope
y = f x
f
y ' = f ' x
given x1
the slope of a tangent line
http://en.wikipedia.org/wiki/Derivative
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Derivatives (1A) 10 Young Won Lim12/29/15
Differential in calculus
x1 x1 dx
f x1
f x1 dx
dx
dy
Differential: dx, dy, … ● infinitesimals● a change in the linearization of a function● of, or relating to differentiation
the linearization of a function
function
http://en.wikipedia.org/wiki/Derivative
f x1
f x1 dx
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Derivatives (1A) 11 Young Won Lim12/29/15
Approximation
x1 x1 dx
f x1
f x1 dx
dx
dy
Differential: dx, dy, …
the linearization of a function
function
http://en.wikipedia.org/wiki/Derivative
f x1
f x1 dx
f (x1 + dx ) ≈ f (x1) + dy
= f (x1) + f ' (x1)dx
f ' (x1) = limh→0
f (x1 + h) − f (x1)
h
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Derivatives (1A) 12 Young Won Lim12/29/15
Differential as a function
x1
f (x1)
Line equation in the new coordinate.
dx
dydy = f ' (x1) dx
slope = f ' (x1)
http://en.wikipedia.org/wiki/Derivative
dx1 dx2 dx3
dy1 dy2 dy3
=dy1
dx1
=dy2
dx2
=dy3
dx3
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Derivatives (1A) 13 Young Won Lim12/29/15
dy = f ' (x) dx
dydx
= f ' (x)
ratio not a ratio
dy =dfdx
dx
Differentials and Derivatives (1)
differentials derivativex1
f (x1)
dx
dydy = f ' (x1) dx
slope = f ' (x1)
http://en.wikipedia.org/wiki/Derivative
dx1 dx2 dx3
dy1 dy2 dy3
=dy1
dx1
=dy2
dx2
=dy3
dx3
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Derivatives (1A) 14 Young Won Lim12/29/15
Differentials and Derivatives (2)
for small enough dx
f (x1 + dx ) ≈ f (x1) + dy
= f (x1) + f ' (x1)dx
f (x1 + dx ) = f (x1) + dy
= f (x1) + f ' (x1)dx
limdx→0
f (x1 + dx ) − f (x1)
dx= f ' (x1)
limdx→0
x1
f (x1)
dx
dydy = f ' (x1) dx
slope = f ' (x1)
http://en.wikipedia.org/wiki/Derivative
dx1 dx2 dx3
dy1 dy2 dy3
=dy1
dx1
=dy2
dx2
=dy3
dx3
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Derivatives (1A) 15 Young Won Lim12/29/15
dy = f ' (x) dx
dy =dfdx
dx
Differentials and Derivatives (3)
dy = f dx
dy = Dx f dx
∫ dy = ∫ dfdx
dx
∫ dy = ∫ f ' (x) dx
∫ dy =∫ 1dy = y
y = f (x)
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Derivatives (1A) 16 Young Won Lim12/29/15
Integration Constant C
∫ dy = ∫ dfdx
dx
∫ dy = ∫ f ' (x) dx
y + C1 = f (x) + C2
y = f (x) + C
place a constant
place another constant
∫ dy = ∫ dfdx
dx + C
differs by a constant
∫ dy = ∫ f ' (x) dx + C
y = f (x) + C
place only one constant from the beginning
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Derivatives (1A) 17 Young Won Lim12/29/15
The differential of a function ƒ(x) of a single real variable x is the function of two independent real variables x and dx given by
(x , dx ) dy
Differential as a function
x1
f x1
Line equation in the new coordinate.
dx
dydy = f ' x1 dx
slope = f ' x1
http://en.wikipedia.org/wiki/Derivative
dy = f ' (x) dx
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Derivatives (1A) 18 Young Won Lim12/29/15
Applications of Differentials (1)
∫ f ( g(x ) )⋅g ' (x) dx ∫ f ( u ) du=
Substitution Rule
u = g( x) du = g ' (x )dx
∫ f (g)dgdx
dx ∫ f ( g ) dg=
du =dgdx
dx(I)
(II)
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Derivatives (1A) 19 Young Won Lim12/29/15
Applications of Differentials (2)
∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=
Integration by parts
u = f (x)
v = g(x )
du = f ' (x ) dx
dv = g ' ( x)dx
∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=
∫udv u v − ∫ v du=
du =dfdx
dx
dv =dgdx
dx
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Derivatives (1A) 20 Young Won Lim12/29/15
Derivatives and Differentials (large dx)
dx = 0.6;
f (x ) = sin(x )
tangent at x1
m = f ' (x1) f (x1) + f ' ( x1)dx
f ' (x1)dx
f (x1) = sin(x1)
dx
f (x2) + f ' (x2)dx
x1 x2
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Derivatives (1A) 21 Young Won Lim12/29/15
Euler Method of Approximation (large dx)
dx = 0.6;
f (x ) = sin(x )
tangent at x1
m = f ' (x1) f (x2) ≈ f (x1) + f ' (x1)dx
f (x1) = sin(x1)
dx
f (x3) ≈ f (x2) + f ' (x2)dx≈ (f (x2) + f ' (x2)dx ) + f ' (x2)dx
x1 x2 x3
Euler Method
Initial Value
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Derivatives (1A) 22 Young Won Lim12/29/15
Derivatives and Differentials (large dx = 0.6)
dx = 0.6;
dy = f ' (x) dx
dy =dfdx
dx ∫ dy = ∫ dfdx
dx
∫ dy = ∫ f ' (x) dx
y = f (x)
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Derivatives (1A) 23 Young Won Lim12/29/15
Derivatives and Differentials (small dx = 0.2)
dx = 0.2;
dy = f ' (x) dx
dy =dfdx
dx ∫ dy = ∫ dfdx
dx
∫ dy = ∫ f ' (x) dx
y = f (x)
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Derivatives (1A) 24 Young Won Lim12/29/15
Euler's Method of Approximation
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Derivatives (1A) 25 Young Won Lim12/29/15
Octave Code
clf; hold off;dx = 0.2;
x = 0 : dx : 8;y = sin(x);plot(x, y);t = sin(x) + cos(x)*dx ;y1 = [y(1), t(1:length(y)-1)];
y2 = [0];y2(1) = y(1);for i=1:length(y)-1 y2(i+1) = y2(i) + cos((i)*dx)*dx;endfor
hold ont = 0:0.01:8;plot(t, sin(t), "color", "blue");plot(x, y, "color", 'blue', "marker", 'o');plot(x, y1, "color", 'red', "marker", '+');plot(x, y2, "color", 'green', "marker", '*');
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Differentiation 26 Young Won Lim12/29/15
Derivatives of inverse trigonometric functions
y = arcsin x
−π2
≤ y < + π2
sin y = x
ddx
(sin y) =ddx
x
=1
√ 1 − x2
y = arccos x
0 ≤ y < + π
cos y = x
ddx
(cos y) =ddx
x
=−1
√ 1 − x2
y = arctan x
−π2 ≤ y < + π
2
tan y = x
ddx ( sin y
cos y ) =ddx
x
=1
1 + x2
(cos y)⋅ y ' = 1
y ' =1
cos y=
1
√ 1 − x 2
d ydx
=ddx
(arcsin x)
(−sin y)⋅ y ' = 1
y ' =−1
sin y=
−1
√ 1 − x2
d ydx
=ddx
(arccos x)
cos2 y⋅ y '+ sin2 y⋅ y '
cos2 y= 1
(1 + tan2 y)⋅ y ' = 1
d ydx
=ddx
(arctan x)
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Young Won Lim12/29/15
References
[1] http://en.wikipedia.org/[2] M.L. Boas, “Mathematical Methods in the Physical Sciences”[3] E. Kreyszig, “Advanced Engineering Mathematics”[4] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”