# Solving systems of equations and derivatives

I am new to Mathematica, and I'm not sure how to solve a system of equations that include differentiation.

Given that f and g are differentiable functions:

f(x) < 0 for all x
g(5) = 2
h(x) = f(x) / g(x)
h'(x) = f'(x) / g(x)
Then g(x) = ?


I didn't expect it to work, but I tried this:

DSolve[{f[x] < 0, g[5] = 2, h[x] == f[x]/g[x], h'[x] = f'[x]/g[x]}, g[x], x]


And of course it didn't work.

How would I do this?

• Possibly a duplicate... – kukac67 Apr 8 '14 at 21:40
• Reduce[D[f[x]/g[x], x] == f'[x]/g[x], g[x]] – Dr. belisarius Apr 8 '14 at 21:54
• @belisarius beat me to this, but...h[x] = f[x]/g[x]; Solve[D[h[x], x] == f'[x]/g[x], g'[x]] – Daniel Lichtblau Apr 8 '14 at 21:56
• @DanielLichtblau Why does that work? – kukac67 Apr 8 '14 at 21:58
• And thanks for the answer! – kukac67 Apr 8 '14 at 22:00

Reduce[ ] tells you something useful about those functions:
Reduce[D[f[x]/g[x], x] == f'[x]/g[x], g[x]]