<s>[INST] Differentiation: $y = 9x + \frac{3}{x}$ Differentiate $y = 9x + \frac{3}{x}$
I think the first step is to turn $\frac{3}{x}$ into a more "friendly" format, so $x$ to the power of something maybe? How do I get $x$ with an index from $\frac{3}{x}$?
 [/INST] You can also apply the definition (whenever $x\neq 0$):
\begin{align*}
f(x) = 9x + \frac{3}{x} \Rightarrow f'(x) & = \lim_{h\to 0}\frac{1}{h}\left(9(x + h) - 9x + \frac{3}{x + h} - \frac{3}{x}\right)\\\\
& = \lim_{h\to 0}\frac{1}{h}\left(9h - \frac{3h}{x(x + h)}\right)\\\\
& = \lim_{h\to 0}\left(9 - \frac{3}{x(x+h)}\right)\\\\
& = 9 - \frac{3}{x^{2}}
\end{align*}
Hopefully this helps !
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