Differentiation of composite function - integral form

(See attached file for full problem description with proper symbols) --- Assume that f is continuous on [a,b], g is differentiable on [c,d], g([c,d]) [a,b] and F(x) = For each x [c,d]. Prove that F’(x)=f(g(x))g’(x) For each x (c,d).

Proof involving integral

(See attached file for full problem description with proper symbols) --- Assume that f is continuous on [a,b], g is differentiable on [c,d], g([c,d]) [a,b] and F(x) = For each x [c,d]. Prove that F’(x)=f(g(x))g’(x) For each x (c,d). ---

Subset of a countable set is countable

Prove that every subset S of a countable set X is itself countable.

Union of countable set

Prove that union of countable sets is countable. See attached file for full problem description.

Determine derivative

(See attached file for full problem description)