Solve this Derivative

By Nick | Published: 17.09.2009

Given:

$f(x)=\frac{x}{\sqrt[]{x} + 1}$

Find:

$\large{f}^\prime\left(4\right)$

Now, I can do this using the quotient rule, but I need to do it without using any “shortcuts”. Using that rule, we get:

$\large{f}^\prime\left(x\right)=\frac{\sqrt[]{x+1}-\frac{x}{2\sqrt[]{x}}}{\left(\sqrt[]{x+1}\right)^2}$

Which you can simplify and get:

$\large{f}^\prime\left(4\right)=\frac{2}{9}$

But without the quotient rule we have this mess:

$\large\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow0}\frac{\frac{x+h}{\sqrt[]{x+h}+1}-\frac{x}{\sqrt[]{x}+1}}{h}$

I’ve tried a few times, but I can’t seem to get it simplified enough.

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