<s>[INST] Using the limit definition to find a derivative for $\frac{-5x}{2+\sqrt{x+3}}$ I am trying to find the derivative of $\frac{-5x}{2+\sqrt{x+3}}$ using the limit definition of a derivative.
$$\lim_{h \to 0} \frac {f(x+h)-f(x)}{h}$$
What I did is
$$\lim_{h \to 0} \frac{\dfrac{-5(x+h)}{2+\sqrt{x+h+3}}\dfrac{2-\sqrt{x+h+3}}{2-\sqrt{x+h+3}}-\dfrac{-5x}{2+\sqrt{x+3}}\dfrac{2-\sqrt{x+3}}{2-\sqrt{x+3}}}{h}$$
But I am having problem finding the correct answer.
 [/INST] Don't rationalize like that; instead consider
\begin{align}
&\lim_{h\to0}\frac{1}{h}\left(
\frac{-5(x+h)}{2+\sqrt{x+h+3}}-\frac{-5x}{2+\sqrt{x+3}}\right)
\\[6px]
&\qquad=
\lim_{h\to0}\frac{
-10x-5x\sqrt{x+3}-10h-5h\sqrt{x+3}+10x+5x\sqrt{x+h+3}
}{h(2+\sqrt{x+h+3})(2+\sqrt{x+3})}
\\[6px]
&\qquad=
\lim_{h\to0}\frac{-10-5\sqrt{x+3}}{(2+\sqrt{x+h+3})(2+\sqrt{x+3})}+
\lim_{h\to0}\frac{5x(\sqrt{x+h+3}-\sqrt{x+3})}{h(2+\sqrt{x+h+3})(2+\sqrt{x+3})}
\\[6px]
&\qquad=\frac{-10-5\sqrt{x+3}}{2(2+\sqrt{x+3})}+
\frac{5x}{2(2+\sqrt{x+3})}\lim_{h\to0}\frac{\sqrt{x+h+3}-\sqrt{x+3}}{h}
\end{align}
and you should be able to finish up.
</s>