Question

Using the definition of the derivative, differentiate the function f(x) = x + 9/x

Answer 1

you mean using first principle?

Answer 2

\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]
Not sure what your given function is, could be
\[f(x)=x+\frac{9}{x}~~\text{ or }~~f(x)=\frac{x+9}{x}\]

Answer 3

no matter which case, just try to plug it in :)

Answer 4

it's the first function you wrote SithsAndGiggles

Answer 5

\[\lim_{h\to0}\frac{\left(x+h+\frac{9}{x+h}\right)-\left(x+\frac{9}{x}\right)}{h}\\
\lim_{h\to0}\frac{h+\frac{9}{x+h}-\frac{9}{x}}{h}\\
\lim_{h\to0}\frac{h}{h}+\lim_{h\to0}\frac{\frac{9}{x+h}-\frac{9}{x}}{h}\\
\lim_{h\to0}\frac{h}{h}+\lim_{h\to0}\frac{\frac{9x-9(x-h)}{x(x+h)}}{h}\\
\lim_{h\to0}\frac{h}{h}+\lim_{h\to0}\frac{\frac{9h}{x(x+h)}}{h}\\
\lim_{h\to0}\frac{h}{h}+\lim_{h\to0}\frac{9}{x(x+h)}\\
~~~~~~~~~~~~~~~~~~~~\vdots
\]

Answer 6

Missing a minus sign on the last term:
\[~~~~~~~~~~~~~~~~~~~~\vdots\\
\lim_{h\to0}\frac{h}{h}\color{red}{-}\lim_{h\to0}\frac{9}{x(x+h)}\\
~~~~~~~~~~~~~~~~~~~~\vdots
\]