I understand how to find keys from Key Signatures but how is Fmaj dervived from the Bb signature?

Not sure I follow you - Fmaj is not "derived" from the Bb signature, it only has a Bb in it. Fmaj has a Bb in it as B is required to be flatted in order to uphold the step pattern of a major scale starting on F (whole, whole, half, whole, whole, whole, half).

An understanding of the "circle of fifths" will help you identify the Flats and sharps in any key and explain why they are where they are. Hopefully, this link will help answer your question:

http://www.folkblues.com/theory/circle_5ths_text.htm

Hi Gilbert,

8ude got the gist of it for you. Here's a slightly expanded explanation.

In Western music, we define a key by it's principle tone (or tonic). If the key is major, it will have seven tones in the following order: 1-1-1/2-1-1-1-1/2, where 1 and 1/2 are the steps between tones beginning with the home or tonic key.

So if you start with C major, which blessedly for us pianists uses all white keys, you can see that from C to D is a whole step, D to E is a whole step, but E to F is a half (1/2) step.

In a major key, the distance between the 7th and 8th tone is always a half step. In C major, B to C. In G major, F# to G. In F major, E to F. And the distance between the 3rd and 4th tone is always a half step. So in C major, E to F, and in F major, A to Bb.

The relationship holds for all 12 major keys. Minors follow similar patterns, but are more complex for this discussion.

Does this help any? I hope so. It's a bit wordy, but then, I'm a teacher and words . . . . sorry. I'll end now.