Differentiation

Differentiation Topic 5

to measure the rate at which a function changes differentiation of function y = f(x) is written as d

Differentiation

f(x) or dx dy or dx f' (x) (read as f prime x)

1

2

Rules of Differentiation

Rules of Differentiation

Rule 1: If y = c, where c is constant then

Rule 2: If y = cx, where c is constant then

dy =0 dx

dy =c dx Example dy y = 4x then =4 dx

Example dy y = 4 then =0 dx

dy =0 dx

y =-5 then

y =-5x then

dy = -5 dx

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Rules of Differentiation

Rules of Differentiation

Rule 3: If y = xn, where n is real number then dy = nx n - 1 dx

Example y = x2 then

y = 2x3 then

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Rule 4: If y = ln u then

,n ≠1

Example dy y = ln 2x then

dy = 2x 2 - 1 = 2x dx

dx

dy = 3.2x 3 - 1 = 6x 2 dx

y = ln 2x3 then 5

dy 1 du = dx u dx

=

1 d 1 1 . 2x = .2 = x 2x dx 2x

dy 1 d 1 3 = . 2x3 = 3 .6x2 = dx 2x3 dx x 2x 6

1

Rules of Differentiation

Rules of Differentiation

Rule 5: dy du If y = eu then = eu .

Rule 6: dy = f' (x) + g' (x) If y = f(x) + g(x) then dx

dx

dx

Example dy d y = ex then = ex . x = e x .1 = e x 3

dx

dx

dx

y = e2x then

Example dy d d = 1 =2 2x + y = 2x + 1 then

dy d = e 2x . 2x 3 = e 2x . 6x 2 dx dx 3

3

y = x + 3x3 then

dx

dx

dy d d = x + 3x 3 = 1 + 9x 2 dx dx dx

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Rules of Differentiation

Rules of Differentiation

Rule 7: dy If y = f(x) - g(x) then = f' (x) - g' (x) dx

Example y = 2x + 1 then y = x + 3x3 then

dy d d = 3x 3 = 1 - 9x 2 xdx dx dx 9

Rules of Differentiation

[g(x)]

Example d d x. 4x 5 - 4x 5. x dx 4x 5 then dy = dx y = dx x2 x 5

=

20x - 4x x

2

5

=

16x x

2

5

dy = f(x). g' (x) + g(x).f' (x) dx

dy d d = 2x. (x 2 + 1) + (x 2 + 1). 2x dx dx dx = 2x(2x) + (x 2 + 1)(2)

= 4x 2 + 2x 2 + 2 = 6x 2 + 2

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Rules of Differentiation

Rule 9: f(x) dy g(x).f'(x) - f(x).g'(x) = If y = then 2 dx

Rule 8: If y = f(x).g(x) then Example y = 2x(x2 + 1) then

dy d d = 1 =2 2x dx dx dx

g(x)

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Rule 10: If y = un where u=f(x) and n=constant then dy n - 1 du dx

=

x.20x4 - 4x 5. 1 x2

= n.u

.

dx

Example y = (2x – 3x2)3 then

dy d = 3.(2x - 3x 2 ) 3 - 1 . (2x - 3x 2 ) dx dx

= 16x3 11

= 3.(2x - 3x 2 ) 2 .2 - 6x

= (6 - 18x)(2x - 3x 2 ) 2

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2

Second Order Differentiation

Second Order Differentiation

Second Order Differentiation is produce when we differentiate a function f(x) twice and denoted by

Example from Rule 8: y = 2x(x2 + 1) then dy d d = 2x. (x 2 + 1) + (x 2 + 1). 2x dx dx dx

d f' (x) or dx 2

d y

= 2x(2x) + (x 2 + 1)(2)

= 4x 2 + 2x 2 + 2 = 6x 2 + 2

or

2

dx f'' (x) (read as f double prime x)

2

d y 13

Partial Differentiation

dx

2

=

d (6x 2 + 2) = 12x dx 14

Partial Differentiation

The partial differentiation of f(x,y) with respect to x is d f(x, y)

Example1: find partial differentiation for f(x, y) = 2x 2 - 3y - 4

The partial differentiation of f(x,y) with respect to y is d f(x, y)

Solution

dx

d d 2x 2 - 3y - 4 = 4x f(x, y) = dx dx

dy

d d 2x 2 - 3y - 4 = - 3 f(x, y) = dy dy 15

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Partial Differentiation Example2: find partial differentiation for 2 f(x, y) = (x - 1)(y + 2 )

Solution

d d d d f(x,y)= (x2 -1)(y +2 ) f(x,y)= (x2 -1)(y +2 ) dy dy dx dx d d d d =(x2 -1) (y+2 )+(y+2 ) (x2 -1 ) =(x2 -1) (y+2 ) +(y+2 ) (x2 -1 ) dx dx dy dy

(x2 -1).0 + (y+2 )(2x) = 2xy+4x =

=

(x2 -1).1

= x2 -1

+ (y+2 ) .0 17

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