Dear person,

INPUT/My Question: log(f'(x)) does NOT Equal Wolfram Alphas Answer ??= OUTPUT: d/dx(log(f'(x))) = (f''(x))/(f'(x))

-TIP, Proof by contradiction using substitution:
When run f(x)=e^x AKA f(x)=Euler-e^x through Left Equation see X Does NOT EQUAL ONE (1) SO log(f'(x)) NOT EQUAL -
d/dx(log(f'(x))) = (f''(x))/(f'(x))

I believe Wolfram Alpha site, or maybe Mathematica on backend, mistakenly moves the innermost dx to outside the outermost parentheses?

Given that e^x is its own integral and derivative, maybe you want to always substitute f(x)=e^(x) to always check your answers?

Best Regards,
Marc Cox

Ps> If you find merit in this, 
    let Stephen Wolfram / Community know please 
    because I love to help. :-)