Derivative of $\frac{1}{\sqrt{x+5}}$

I'm trying to find the derivative of $\dfrac{1}{\sqrt{x+5}}$ using $\displaystyle \lim_{h\to 0} \frac {f(x+h)-f(x)}{h}$

So,

\begin{align*} \lim_{h\to 0} \frac{\dfrac{1}{\sqrt{x+h+5}}-\dfrac{1}{\sqrt{x+5}}}{h} &= \frac{\dfrac{\sqrt{x+5}-\sqrt{x+h+5}}{(\sqrt{x+h+5})(\sqrt{x+5})}}{h}\\\\ &= \frac{\dfrac{x+5-x-h-5}{(\sqrt{x+h+5})(\sqrt{x+5})}}{\dfrac{h}{\sqrt{x+5}}+\dfrac{h}{\sqrt{x+h+5}}} \end{align*}

• You have a typo in the first nominator, it should not be $\frac1{\sqrt x}$. – Asaf Karagila Oct 4 '12 at 23:35
• @AsafKaragila: Fixed, thanks! – Austin Broussard Oct 4 '12 at 23:36

We can introduce a variable, say $u:=x+5$. Then it's \begin{align*} \lim_{h\to 0} \frac1h\cdot \left( \frac1{\sqrt{u+h}} -\frac1{\sqrt{u}} \right) &= \lim_{h\to 0} \frac1h\cdot\left(\frac{\sqrt u-\sqrt{u+h}}{\sqrt{u(u+h)}} \right) = \\ &= \lim_{h\to 0} \frac1h\cdot\left(\frac{\sqrt u-\sqrt{u+h}}{\sqrt{u(u+h)}} \cdot \frac{\sqrt{u}+\sqrt{u+h}}{\sqrt{u}+\sqrt{u+h}} \right) = \\ &= \lim_{h\to 0} \frac1h\cdot\left(\frac{u-(u+h)}{\sqrt{u(u+h)}\left(\sqrt{u+h}+\sqrt u\right)}\right) = \\ &=\lim_{h\to 0}\frac{-1}{\sqrt{u(u+h)}\left(\sqrt{u+h}+\sqrt u\right)} =\\ &=\frac{-1}{u\cdot 2\sqrt u} \end{align*} Finally, rewrite back $u=x+5$.
• It's a lot clearer with substitution and taking out $\frac{1}{h}$. Thank you. – Austin Broussard Oct 4 '12 at 23:59
Multiply the numerator and denominator by $\sqrt{x+5}\cdot\sqrt{x+h+5}$ and simplify. Equivalently, put the denominator over a common denominator and simplify the resulting four-story fraction; it’s exactly the same thing. It should also be a fairly automatic response to an expression like this, since it’s the most likely route to a simplification.
Do you know the derivative of logarithms? $$f(x)=\frac{1}{\sqrt{x+5}}\\ Lf(x)=\log f(x)=-\frac{1}{2}\log(x+5)\\ \frac{f'(x)}{f(x)}=\bigg( -\frac{1}{2}\log(x+5) \bigg)'_x\\ f'(x)=f(x)\bigg( -\frac{1}{2}\log(x+5) \bigg)'_x\\ L_1=\lim_{h \to 0}\frac{-\frac{1}{2}\log(x+5+h)+\frac{1}{2}\log(x+5)}{h}=-\frac{1}{2}\lim_{h \to 0}\frac{\frac{1}{2}\log(x+5+h)-\frac{1}{2}\log(x+5)}{h}$$ Derivative of log function is available, e.g., here. Hence, $$L_1=-\frac{1}{2(x+5)}$$ And, therefore $$f'(x)=f(x)L_1=-\frac{1}{2(x+5)^{\frac{3}{2}}}$$