Final answer:
To obtain the focal length formula using object distance and magnification, combine the lens formula with the magnification formula, rearrange, and solve for focal length, resulting in f = do/(m - 1).
Step-by-step explanation:
To derive the formula for the focal length of a lens in terms of the object distance (u) and the magnification (m), we combine the lens formula with the magnification formula. The lens formula is 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance. The magnification formula is m = -di/do.
By rearranging the magnification formula to find di, we get di = -m × do. We can then substitute di in the lens formula, resulting in 1/f = 1/do - 1/(m × do). With further algebraic manipulation, we can solve for f to get the formula f = do/(m - 1), assuming m is not equal to 1.