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differentiation techniques
sachin tendulkar
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Posted on
9 Apr 2009 19:58:58 IST
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9 Apr 2009 19:58:58 IST
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differentiation techniques
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Differential Calculus
Maybe the easiest and most useful formulas are the ones that say that the derivative is linear:
 
egin{displaymath}egin{array}{ll}(f + g)'(a) = f'(a) + g'(a)\(cf)'(a) = c f'(a)\end{array}end{displaymath}

 
Combined with the formula (xn)' = n xn-1, we see that every polynomial function has a derivative at any point.
Example. For P(x) = 1-2x + 3x4 -5 x6, we have
 
egin{displaymath}P'(x) = -2 + 12 x^3 - 30 x^5cdotend{displaymath}

 
The next two formulas are the most powerful ones. They deal with the derivative of a product and a quotient. They are commonly called the product rule and the quotient rule. We have

 
egin{displaymath}egin{array}{lcl}Big(f;gBig)'(a) &=& f'(a)g(a) + f(a) g'......splaystyle rac{f'(a)g(a) - f(a) g'(a)}{g^2(a)}\end{array}end{displaymath}

 
In particular, we have
 
egin{displaymath}left( rac{1}{f}ight)'(a) = - rac{f'(a)}{f^2(a)}cdotend{displaymath}

 
So, we have
 
egin{displaymath}left( rac{1}{x^n}ight)' = - rac{nx^{n-1}}{x^{2n}} = -n x^{-n-1}end{displaymath}

 
which means that the formula (xr)' = r xr-1 is also valid for negative exponents.
Before we discuss the derivative of trigonometric functions, let us stop here and reflect a little bit more on polynomial functions. Indeed, we saw that the derivative of a polynomial function is also a polynomial function. So we can take another derivative and generate a new function. This function is called the second derivative. We can keep doing this as long as we want to. The functions obtained are called higher derivatives. The common notations used for them are
 
egin{displaymath}egin{array}{llllllll}f'(x), &; f''(x), &; f'''(x), &; f...... &; cdots, &;displaystyle rac{d^ny}{dx^n} \end{array}end{displaymath}rate this if you like

Comments (2)
mahesh raja
New kid on the Block

Joined: 9 Apr 2009 12:01:54 IST
Posts: 10
9 Apr 2009 20:24:24 IST
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nice!!!!!!!!!!!!!!!!!!!!!!!!!!

kishore
New kid on the Block

Joined: 16 Mar 2009 11:20:16 IST
Posts: 23
9 Apr 2009 20:25:55 IST
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wow! good work chap


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