Question

Find the derivative of f(x) = 7 divided by x at x = 1

Answer 1

If you know the rules for differentiation, then

`f(x)=\frac7x\implies f'(x)=-\frac7{x^2}\implies f'(1)=-7`

If you don't, you can use the definition of the derivative to first find
`f'(x)`:

`f'(x)=\displaystyle\lim_(h\to0)\frac{\frac7{x+h}-\frac7x}h=\lim_(h\to0)(7x-7(x+h))/(hx(x+h))=-\lim_(h\to0)\frac7{x(x+h)}=-\frac7{x^2}`

Plug in
`x=1` to get the same result.

Or, you can directly compute the value of the derivative using the limit definition:

`f'(1)=\displaystyle\lim_(x\to1)(\frac7x-7)/(x-1)=\lim_(x\to1)(7(1-x))/(x(x-1))=-\lim_(x\to1)\frac7x=-7`