Up until now, I've been finding tangent slope functions I've just been using this with formula with the given function:
$\frac{f(x+h)-f(x)}{h}$
Which have worked fine so far for regular quadratic and cubic functions. However, I'm not sure how to do it with a reciprocal function, for example $y=\frac{1}{x-1}$
When I try, I just end up doing this:
$\frac{\frac{1}{(x+h)-1}-\frac{1}{x-1}}{h}$
And from there I have no idea how to derive the function into one with only integers, x's and h's, and no fractions (which with I let $h=0$ to find the tangent slope function).
How can I find the tangent slope function for these types of functions?
Best Answer
In your example (which, btw, has a mistake in your last mathematical expression...):
$$\frac{\frac1{x+h-1}-\frac1{x-1}}{h}=\frac{-\color{red}h}{\color{red}h(x+h-1)(x-1)}\xrightarrow[h\to 0]{}\frac{-1}{(x-1)^2}$$