Question

Sally is differentiating a function `f`. She has gotten the answer, `f'(x) = −x^-2`. Write 3 possibilities for the function which Sally differentiated?

Answer 1

I get only two possibilities
1. `f(x)=1/x+C_1`
2. `f(x)=1/x`

Explanation:

Although it appears from the text to be a question on differentiation. However, if you look closely it is actually a question of integration.

Given `f'(x)=-x^-2`, need to find out `f(x)` and state three possibilities.
Integrating both sides we get
`intf'(x)dx=int-x^-2 dx`
`=>f(x)=-int1/x^2 dx`

Using the basic form
`intx^n dx=1/(n+1)x^(n+1)`
we get

`f(x)=-(-1/x+C)`
where `C` is a constant of integration.
`=>f(x)=1/x+C_1`
where `C_1` is a constant of integration.
I get only two possibilities
1. `f(x)=1/x+C_1`
2. `f(x)=1/x`