Given $\displaystyle f(x)=\frac{1}{x} $.Find $\displaystyle \frac{f(x+h)+f(x)}{h} $ where h is not equal to zero
Attempt:
$\displaystyle \frac{\frac{h}{x+h*x)}}{h} $
How to simplify (please provide steps)
Thank you
$\displaystyle f(x) = \frac{1}{x}$
$\displaystyle f(x + h) = \frac{1}{x + h}$
$\displaystyle f(x + h) - f(x) = \frac{1}{x + h} - \frac{1}{x}$
$\displaystyle = \frac{x}{x(x + h)} - \frac{x + h}{x(x + h)}$
$\displaystyle = -\frac{h}{x(x + h)}$
$\displaystyle \frac{f(x +h) - f(x)}{h} = \frac{-\frac{h}{x(x + h)}}{h}$
$\displaystyle = -\frac{h}{x(x + h)}\cdot\frac{1}{h}$
$\displaystyle = -\frac{1}{x(x + h)}$.
To learn, in general, how to evaluate a function at an expression (rather than a number), try here.
Then try working this exercise in pieces, rather than all at one. First write down the formula for f(x). Then find the expression for f(x + h). Then subtract the polynomial for f(x) from the polynomial for f(x + h). Then divide by h, cancelling if you can.