Let g and p be differentiable functions on an open interval I and suppose that g(x)p(x)=k(x) for all x in I.
Explain, in a clear sequence of complete sentences, how one can obtain a general expression for p′(x) that depends only on g(x), g′(x), k(x), and k′(x). Your explanation should:
• identify the fundamental relationship between the derivatives involved,
• justify each algebraic manipulation used to isolate p′(x), and
• comment on any conditions that must be satisfied for the final expression to be valid (for example, restrictions on g).
Do not perform any numerical calculations; focus solely on the logical development of the formula.