Difference of Quotient

Question

What is the difference of quotient when f(x)=1/√x? My teacher mentioned we would have to rationalize it and I know it would then be √x/x, so what would be the answer? I keep getting lost on the steps.

Answer 1

1/rad(x+h) - 1/rad(x)=

rad(x) - rad(x+h)

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rad(x+h)rad(x)

rationalizes numerator:

[rad(x) - rad(x+h)] [rad(x)+rad(x+h)]

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rad(x+h)rad(x) [ rad(x)+rad(x+h)]

x - (x+h)

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x rad(x+h) + (x+h)rad(x)

numerator is -h which gets divided by h from the difference quotient

-1

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

x rad(x+h) + (x+h)rad(x)

limit as h-->0

-1

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2 x^(3/2)

This agrees with the power rule Dx [ x^(-1/2) = -1/2( x^(-3/2))