Question

How to differentiate this function to find f'(7)?

Suppose that f(x) is a differentiable function such that f(7+h)=2/(h+1) and that f(7)=2. Then f'(7) = ???
The answer is -2, I'm just not sure how to reach it.

Answer 1

See below

Explanation:

We use the definition of the derivative

`f^'(x) = lim_{h to 0}(f(x+h)-f(x))/h`

For the given function, we have

`f^'(7) = lim_{h to 0} (f(7+h)-f(7))/h`
`qquad = lim_{h to 0}1/h(2/(h+1)-2)`
`qquad = lim_{h to 0}1/h(2-2h-2)/(h+1)`
`qquad = lim_{h to 0}-2/(h+1)`
`qquad = -2`

Alternative

You could also simply recognize that the function is simply

`f(x) = 2/((x-7)+1) = 2/(x-6)`

so that

`f^'(x) = -2/(x-6)^2 implies f^'(7) = -2`

I guess from the way the problem is worded, though, that a calculation using the definition of the derivative is really being sought here.