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# Difference Quotient

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Been stuck on this one for a while. Help appreciated!

Jun 3, 2021

#1
+121085
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sqrt  [( x + h)^2  + 9 ]   -  sqrt  [ x^2  + 9]

_________________________________

h

sqrt  [  x^2  + 2xh  + h^2   +  9 ]  -  sqrt  [ x^2 + 9]

__________________________________________

h

Multiply top / bottom  by    sqrt  [ x^2  + 2xh  + h^2 + 9 ]  +  sqrt [  x^2 + 9 ]

x^2  + 2xh   + h^2  + 9   - x^2 - 9

________________________________________     =

h   [ ( sqrt x^2 + 2xh + h^2 + 9)  +  (sqrt (x^2 + 9) ]

2xh  + h^2

______________________________________    =

h   [ ( sqrt x^2 + 2xh + h^2 + 9)  +  (sqrt (x^2 + 9)  ]

h (2x + h)

_______________________________________  =

h   [ ( sqrt x^2 + 2xh + h^2 + 9)  +  (sqrt (x^2 + 9)  ]

2x  +  h

______________________________________           let   h approach  0     and we have

[ ( sqrt x^2 + 2xh + h^2  + 9)  +  (sqrt (x^2 + 9)  ]

2x

_______________________________________      =

sqrt (x^2 + 9)   +  sqrt (x ^2  + 9)

2x

_________________________________  =

2sqrt (x^2 + 9)

x

____________________

sqrt (x^2 + 9)

If you take Calculus.....you  will   learn  that   the  derivative of  (x^2 + 9)^(1/2)  =

(1/2)  (2x)  / ( (x^2 + 9)^(1/2)  =       x   /  sqrt ( x^2 + 9)        =   our result  !!!!

Jun 3, 2021