Do you remember ODE
Ordinary Differentiate Equation?
Do you like to be reminded of
How much a fine mathematician
love to work on math as if one is a magician?
Let me teach you
about how to differentiate a polynomial
By the power rule.
The rate of change in distances at certain time period is called speed,
The rate of change in the increase of population is called average growth,
How about the rate of change in each second?
The way to calculate the instant rate of change
Is to differentiate…
For a polynomial,
The derivatives to any fixed number or constant
Is zero,
The derivative to 1 is zero,
The derivative to 2 is again zero,
The derivative to 200 is also zero..
What a hero.
The derivatives to any power of x
Is to bring down the power as a coefficient,
Reduce the power to x by one.
For example,
The derivative of x to the power 5
Is 5 times x to the power 4…
Do you get it?
Did you picture it?
The derivative of x to the power 2
Is 2 times x to the power 1,
Which is the same as 2 times x or 2x,
How divine!
If you stick to the rule,
You can do it correct and feel cool.
Keep working on it,
Keep thinking of it,
You will make it,
…
Thanks to Trisha, Fiveloaf, and Pravin who have nominated Jingle for The PERFECT Poet Award , which Jingle feels very honored and accepts here, Please visit them to read their OUTSTANDING work in poetry today…
Trisha
Fiveloaf
Pravin nair
For Week 27 The Perfect Poet Award, Jingle wishes to nominate (she is nominated by 3 poets, thus decides to nominate three poets here.)
Tracyhsays
Leonnyes
THE BEATY
Congratulations! Tracy, Leonnyes, and JP, I enjoy your talent in writing or poetry and feel very blessed to have you join Poets Rally Week 26…
*****
I wold like to make an announcement about Pravin Nair whose poetry has been published as a book!
The link to Pravin Nair’s book publishing announcement:
http://www.versepoems.com/2010/08/andmybookisborn.html
Let’s Celebrate by visiting her and give her the greetings and love she deserves! cheers!
*****
The Power Rule
“The derivative of a power of x
is equal to the product of
the exponent times
x with the exponent reduced by 1.”
That is called the power rule. For example,
However, we know that the power rule is true when n = 1:


d
dx 
x^{1} 
= 1· x^{0} = 1; 

that it is true when n = 2: 



d
dx 
x^{2} 
= 2x; 

and that it is true when n = 3 : 



d
dx 
x^{3} 
= 3x². 
It seems natural, then, to give a proof by induction; The induction hypothesis will be that the power rule is true for n = k:
and we must show that it is true for n = k + 1; i.e. that
d
dx 
x^{k+1} 
= (k + 1) x^{k}. 
Now,
d
dx 
x^{k+1} 
= 
d
dx 
x· x^{k} 
d
dx 
x^{k+1} 
= 
x· k x^{k−1} + x^{k}· 1, 

d
dx 
x^{k+1} 
= 
k x^{k} + x^{k} 

d
dx 
x^{k+1} 
= 
(k + 1)x^{k}. 
Therefore, if the power rule is true for n = k, then it is also true for its succesor, k + 1. And since the rule is true for n = 1, it is therefore true for every natural number.
Problem. Calculate the derivative of x^{6} − 3x^{4} + 5x^{3} − x + 4.
6x^{5} − 12x^{3} + 15x² − 1