Question

Is `f(x)=e^x/sqrt(x^2-x)` increasing or decreasing at `x=3`?

Answer 1

Calculate the derivative via the quotient rule:
`f'(x)=(e^xsqrt(x^2-x)-e^x(2x-1)/(2sqrt(x^2-x)))/(x^2-x)`
`=e^x(x^2-2x+1/2)/(x^2-x)^(3/2)`

If this is positive, then `f` is increasing.

`f'(3)=e^3(9-6+1/2)/(9-3)^(3/2)=7/2e^3 6^(3/2)>0`, because each of its component multiplicative parts is `>0`. So `f(x)` is increasing at `x=3`.

For a sanity check, compare the graph of the function:
graph{(e^x)/sqrt(x^2-x) [-9.26, 14.7, 2, 13.99]}