Simplify the fractional expression

Question

1/(x+h)^2 - 1/(x)^2 / h

Answer 1

[1/(x+h)² - 1/x²] / h

 

= { x²/[x²(x+h)]² - (x+h)²/[x²(x+h)²] } / h

 

= [x² - (x+h)²] / [hx²(x+h)²]

 

= [x² - (x²+2hx+h²)] / [hx²(x+h)²]

 

= -h(2x+h) / [hx²(x+h)²]

 

= (-2x-h) / [x²(x+h)²]

 

= (-2x-h) / [x²(x²+2hx+h²)]

 

= (-2x-h) / (x⁴+2hx³+h²x²) ← Simplified Expression

 

The meaning of this expression is the average rate of change of the function f(x) = 1/x² in the interval [x,x+h].

The limit h→0 is easier to compute, since you can just plug in h=0. Such a limit is it's instantaneous rate of change or the derivative, d/dx 1/x² which is (-2x-0)/(x⁴+2·0x³+0²x²) = -2/x³.