Given f(x)=(1/x^2), find (f(x+h)-f(x))/h

Question

H is to be replaced with (delta x)

Answer 1

f(x + h) = 1/(x+h)^2 = 1/(x^2 + 2xh + h^2)

 

So the numerator (top) is :

 

1/(x^2 + 2xh + h^2) - 1/x^2 

 

  =  x^2 - (x^2 + 2xh + h^2)                     - 2xh - h^2

  ----------------------------------- =   --------------------------------

          x^2 (x^2 + 2xh + h^2)                 x^2 (x^2 + 2xh + h^2)         

 

 

Dividing by h results in:

 

- 2x - h

--------------

x^2 (x^2 + 2xh + h^2)

 

 

 

 

What happens next is h = delta x approaches zero and vanishes.

So it disappears in the numerator. In the denominator, it kills the 2x and h^2 goes away.

So after you take the limit as h-->0, all that is left is

 

-2x

-----

x^4

 

which simplifies to  -2x^(-3)

 

This result agrees with the power rule for finding the first derivative.

 

Preview of coming attractions