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Unformatted text preview: . sin (∆ x)} − sin(x)
∆x δ f(x)
{sin(x) . cos(δ x) + cos (x) . sin (δ x)} − sin(x)
=
δx
δx
When x is infinitely small, say δx , then as we said earlier, we may let : 2. TENDS TO step: sin ( δx) = δx and cos ( δx) = 1
δ f(x)
=
δx
= {sin(x) . 1+ cos (x) . δ x} − sin(x)
δx
cos (x) . δ x
δx Here we may simplify since δx ≠ 0 : δ f(x)
= cos(x)
δx 3. LIMIT step: Finally we take the LIMIT as δx → 0 .
df(x)
LIMIT δf
LIMIT
= δx→ 0 δx = δx→ 0 { cos (x)}
dx
f’(x) = cos(x) 76 6:
differentia
Example 6 Let us now differentiate the function y = sin (θ) the Vedic way.
dif entiate
+1 y y +1 y2 = sin θ2 y1 = sin θ1
θ1 −1 x 1 +1 x θ2 −1 −1 x2 +1 x −1 Let us now magnify the two diagrams and superimpose them.
+1 y
y2 = sin θ2 } y −y
2 1 = sin θ2 − sin θ1 θ
2− θ
1 y1 = sin θ1 ) x2 −1 x 1 +1 x −1 With the concept of arc length = r θ , we can see from the diagram that :
x2 (θ2 − θ1) ≤ y2 − y1 ≤ x1 (θ2 − θ1)
Since (θ2 − θ1) ≠ 0 we may divide throughout by (θ2 − θ1) .
AVERAGE RATE OF CHANGE of y with respect to θ : 77 x2 ≤ y2 − y1
≤ x1
θ2 − θ1 Now we may take the LIMIT as θ2 → θ1 to get the INSTANTANEOUS RATE OF
CHANGE of y with respect to θ :
y2 − y1
L imit x
≤
θ1 θ2 − θ1
θ2 → θ1 1 L imit x
L imit
2≤
→
→ θ2 θ1 θ2 As θ2 → θ1 we can see that x2 → x1 :
dy
x1 ≤  at θ1 ≤ x1
dθ
dy
d
So :
 at θ1 =  (sin θ) at θ1 = x1 = cos θ at θ1.
dθ
dθ
We may drop the subscript of particular angle θ1 to generalise and get :
d
 (sin θ) = cos θ
dθ
There is no need to DIFFERENTIATE a function from FIRST PRINCIPLES each and
every time. We may differentiate the functions that we encounter most frequently
and build a table. Then all we have to do is look up the TABLE OF DERIVATIVES. 78 16. Rules
Tables and Rules A function f(x) expresses CHANGE. The CHANGE could be in position, velocity,
acceleration, temperature, length, area, volume, pressure, charge o...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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