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Intergration

If y³ = x, how would you differentiate this with respect to x? There are three ways:

Method 1

Rewrite it as y = x^(1/3) and differentiate as normal (in harder cases, this is not possible!)

Method 2

Find dx/dy:
dx  =  3y²dy

And now use the fact:	dy	=	1
	dx		dx/dy

So we get:
dy  =  1
dx     3y²
Method 3 Differentiate term by term and use the chain rule:
y³   =   x

d	(y3)	=	d	(x)
dx			dx

The right hand side of this equation is 1, since the derivative of x is 1. However, to work out the left hand side we must use the chain rule.
The left hand side becomes:
d (y³) ×  dy
dy         dx
(although it is not strictly correct to do so, at this level you can think of dy/dx as a fraction in the chain rule. In the line above, imagine that you can cancel the "dy" s, leaving d/dx and y³, which is what we had in the previous line).
Therefore, 3y² × dy  = 1
                         dx
So   dy  =  1
       dx    3y²In this example, method (2) is probably the easiest. However, there are cases when the only possible method is (3).

Example

Differentiate x² + y² = 3x, with respect to x.

d	(x2)	+	d	(y2)	=	d	(3x)
dx			dx			dx

2x + d (y²)×dy  =  3
       dy        dx
2x + 2y dy  = 3
           dx
dy  =  3 - 2x
dx        2y

Example

Differentiate a^x with respect to x.
You might be tempted to write xa^x-1  as the answer. This is wrong. That would be the answer if we were differentiating with respect to a not x.
Put y = a^x .
Then, taking logarithms of both sides, we get:
ln y = ln (a^x)
so ln y = x lna
So, differentiating implicitly, we get: (1/y) (dy/dx) = lna
and so dy/dx = y lna = a^x lna
(copy)  ::D
jom blajar..

My Love Song (simple plan)

Hey Dad look at me
Think back and talk to me
Did I grow up according
To plan?
Do you think I’m wasting
My time doing things I
Wanna do?
But it hurts when you
Disapprove all along


And now I try hard to make it
I just want to make you proud
I’m never gonna be good
Enough for you
I can’t pretend that
I’m alright
And you can’t change me


‘Cuz we lost it all
Nothing lasts forever
I’m sorry
I can’t be Perfect
Now it’s just too late
And we can’t go back
I’m sorry
I can’t be Perfect


I try not to think
About the pain I feel inside
Did you know you used to be
My hero?
All the days
You spent with me
Now seem so far away
And it feels like you don’t
Care anymore


And now I try hard to make it
I just want to make you proud
I’m never gonna be good
Enough for you
I can’t stand another fight
And nothing’ alright


‘Cuz we lost it all
Nothing lasts forever
I’m sorry
I can’t be Perfect
Now it’s just too late
And we can’t go back
I’m sorry
I can’t be Perfect


Nothing’s gonna change
The things that you said
Nothing’s gonna make this
Right again
Please don’t turn your back
I can’t believe it’s hard
Just to talk to you
But you don’t understand


‘Cuz we lost it all
Nothing lasts forever
I’m sorry
I can’t be Perfect
Now it’s just too late
And we can’t go back
I’m sorry
I can’t be Perfect


‘Cuz we lost it all
Nothing lasts forever
I’m sorry
I can’t be Perfect
Now it’s just too late
And we can’t go back
I’m sorry
I can’t be Perfect

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