Question

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

Answer 1

If you know the rules for differentiation, then

$f(x)=\dfrac7x\implies f'(x)=-\dfrac7{x^2}\implies f'(1)=-7$

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

$f'(x)=\displaystyle\lim_{h\to0}\frac{\frac7{x+h}-\frac7x}h=\lim_{h\to0}\frac{7x-7(x+h)}{hx(x+h)}=-\lim_{h\to0}\frac7{x(x+h)}=-\dfrac7{x^2}$

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

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

$f'(1)=\displaystyle\lim_{x\to1}\frac{\frac7x-7}{x-1}=\lim_{x\to1}\frac{7(1-x)}{x(x-1)}=-\lim_{x\to1}\frac7x=-7$