Derivatives (1A) - Wikimedia · 2014. 6. 23. · Derivatives (1A) 19 Young Won Lim 6/23/14 The...

Young Won Lim6/23/14

Derivatives (1A)


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Derivatives (1A) 3 Young Won Lim6/23/14

A triangle and its slope

y = f x

f x1 h − f x1h

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)


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 x2+h2 x2+h1 x2+h

(x1+h , f (x1+h))

h2h1

h

(x1, f (x1))

(x1+h , f (x1+h))



The limit of triangles and their slopes

y = f x

f ' (x1) = limh→0

f (x1 + h) − f (x1)h

x1

f

The derivative of the function f at x1

function f



The derivative as a function

y = f x x1x2x3

f x1

f x2

f x3

f x

x1x2x3

f ' x 1

f ' x 2

f ' x 3

f ' x

= limh→0

f (x + h) − f (x)h

y ' = f ' (x)

Derivative Function


The notations of derivative functions

x1x2x3

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

ẏ = ḟ x Newton's Notation

D x y = D x f x Euler's Notation

not a ratio.

x is an independent variable→ Derivative with respect to x


Another kind of triangles and their slopes

y = f x

f

y ' = f ' x given x1

the slope of a tangent line



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



Differential as a function

x1

f x1

Line equation in the new coordinate.

dx

dydy = f ' x1 dx

slope = f ' x1



Differentiation & Integration of sinusoidal functions

f x = sin x dd x

f x = cos x leads

f x = cos xdd x

f x = −sin x leads

f x = sin x ∫ f x dx = −cos x C lags

f x = cos x∫ f x dx = sin x C lags


Derivative of sin(x)

f (x) = sin(x)

+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope

dd x

f (x ) = cos(x )

leads


Plot of F(x,y)=cos(x)

+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope

F (x , f (x )) = f ' (x) (x i , y i) f ' (x i)

slope=+1

F (x , y)

+1

0

-1

0

+1

0

-1

0

+1

0

-1

0

f (x ) = sin(x)

f ' (x ) = cos(x)


Derivative of cos(x)

f x = cos x

0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope

dd x

f x = −sin x

leads


Plot of F(x,y)=-sin(x)

F (x , f (x )) = f ' (x) (x i , y i) f ' (x i)

slope=+1

F (x , y)

+1

-1

+1

-1

+1

-1

0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope

0 0 0 0 0 0

f (x ) = cos(x)

f ' (x ) = −sin (x)


Integral of sin(x)

f x = sin x

0 1 2 1 0 1 2 1 0 1 2 1 area + 0

∫ f x dx = −cos x C

-1 0 +1 0 -1 0 +1 0 -1 0 +1 0 area - 1

∫0/2

sin t d t = 1

C = 0

C = 1 ∫0xsin (t) d t + 0

∫0xsin (t) d t − 1

= − cos x lags


Integral of cos(x)

f x = cos x

0 1 0 -1 0 1 0 -1 0 1 0 -1 area

∫ f x dx = sin x C

∫0/2

cos x d x = 1

lags

∫0xcost d t

= + sin(x)


The Euler constant e

dd x

ax = limh0

a xh − ax

h= a x lim

h0

ah − 1h

limh→0

ah − 1h

= 1 f ' 0 = 1

dd x

ex = e x

f x = e x f ' x = ex f ' ' x = ex

limh→0

ah − a0

h − 0= 1

e = 2.71828⋯

⇒ a x such a, we call e


The Euler constant e

f ' 0 = 1

dd x

ex = e x

f x = e x

f ' x = ex

f ' ' x = ex

limh→0

ah − a0

h − 0= 1

e = 2.71828⋯

limh→0

ah − 1h

= 1 iif a = e

Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red).



Derivative Product and Quotient Rule

f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)dd x

ddx

( f g) = d fdxg + f d g

dx

f (x )g( x)

dd x

ddx ( fg ) = ( d fdx g − f d gdx ) / g2

f ' (x )g(x ) − f (x)g ' (x )g2(x )


Integration by parts (1)

f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)dd x

ddx

( f g) = d fdxg + f d g

dx

f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)

f g = ∫ f ' g dx + ∫ f g ' dx

∫⋅d x

∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=


Derivative of Inverse Functions

y = f (x )

a

b (a, b) = (a , f (a))

b

a(b, a) = (b , f−1(b))

y = f−1(x )= g(x )

g' (b) = 1f ' (a)

m2 = g' (b)m1 = f ' (a)

= 1f ' (g(b))

= (b ,g(b))

m1m2 = f ' (a)g ' (b)= 1


Derivative of ln(x) (1)

y = f (x )

a

b (a, b) = (a , f (a))

b

a(b, a) = (b , f−1(b))

y = f−1(x )= g(x )

g' (b) = 1f ' (a) m = g' (b)m = f ' (a )

= 1f ' (g(b))

= (b ,g(b))

f (x )= ex g(x ) = ln x = f −1(x)

g' (x )= 1f ' (g(x))

= 1eg(x )

= 1e ln x

= 1x


Derivative of ln |x|

y = ln x

e y = x

ddx

e y = ddx

x

e y dydx

= 1

dydx

= 1e y

= 1x

ddx

ln x = 1x

x>0

ddx

ln(−x )= 1(−x )

⋅(−1) = 1x

x


Differential Equation

f (x ) = e3 x

f ' (x ) = 3e3 x

f ' (x ) − 3 f (x) = 3 e3x−3 e3 x = 0 f ' (x ) − 3 f (x) = 0

f (x ) ?


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”

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