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December 27th, 2015, 11:02 AM   #1
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Derivative from definition

Hello

i have a little problem with this Deriverative from definition



do i correctly arranged it? cause i have some problem with with the result it comes 0 on numerator and good value on denominator
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December 27th, 2015, 11:45 AM   #2
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$$\begin{aligned}
\lim _{x \to x_0}{{1 \over \sqrt x} - {1 \over \sqrt x_0} \over x- x_0} &= \lim_{h \to 0} {{1 \over \sqrt{x_0+h}}-{1 \over \sqrt x_0} \over h} \\
&= \lim_{h \to 0} \frac1h \cdot {\sqrt x_0 - \sqrt{x_0+h} \over \sqrt{x_0+h}\sqrt x_0} \\
&= \lim_{h \to 0} \frac1h \cdot {\sqrt x_0 - \sqrt{x_0+h} \over \sqrt{x_0+h}\sqrt x_0} \cdot {\sqrt x_0 + \sqrt{x_0+h} \over \sqrt x_0 + \sqrt{x_0+h} } \\
&= \lim_{h \to 0} \frac1h \cdot {x_0 - (x_0+h) \over \sqrt{x_0+h}\sqrt x_0 \left(\sqrt x_0 + \sqrt{x_0+h} \right) } \\
&= \lim_{h \to 0} \frac1h \cdot {-h \over \sqrt{x_0+h}\sqrt x_0 \left(\sqrt x_0 + \sqrt{x_0+h} \right) } \\
&= \lim_{h \to 0} {-1 \over \sqrt{x_0+h}\sqrt x_0 \left(\sqrt x_0 + \sqrt{x_0+h} \right) } \\
&= -{1 \over \sqrt x_0\sqrt x_0 \left(\sqrt x_0 + \sqrt x_0 \right) } \\
&= - {1 \over x_0 \cdot 2\sqrt x_0 } \\
&= -{1 \over 2 x_0^\frac32}
\end{aligned}$$
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January 23rd, 2016, 06:25 AM   #3
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Quote:
Originally Posted by chrisplease View Post
Hello

i have a little problem with this Deriverative from definition



do i correctly arranged it? cause i have some problem with with the result it comes 0 on numerator and good value on denominator
I don't know what you mean by a "good value on denominator". If you take the limit of numerator and denominator separately, you should get 0 for both. Instead "rationalize the numerator" by multiplying both numerator and denominator by $\displaystyle \frac{\sqrt{x_0}+ \sqrt{x_0+ h}}{\sqrt{x_0}+ \sqrt{x_0+ h}}$ as v8archie did. (His "h" is $\displaystyle x- x_0$.)
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