I need help with this please

Theorem 1- Let f be a continuous function on the open 2 of R x E x A where A and E are euclidean spaces. Suppose that Daf is continuous on n. Then for a fixed A, the solution 4 = 4(t, to, xo, A) of x' = f(t, x, 1) x(to) = x0 is unique and admits partial derivative D34 with respect to Co. Even more, the map (t, to, xo, )) - D34(t, to, xo, A) is continuous on its domain D = [(t, to, xo, A); ((t, x0, )) En, w_(to, 0, )) )] and x(t) = D34(t, to, xo, )) . ek = ark ' of for all 1 ) 1) Consider the problem x" = -gsin(x) with x(to) = X0; x'(to) = vo. Show that the solution p(t, to, Xo, vo, g) is defined on all R5 and is of class Coo 2) Using the hypothesis of theorem 1 show that Day, that is, ato , 24 exists and is continuous on D and that: y(t) = ato - 24 (t, to, xo, A) is solution of: y' = J(t)y with y(to) = -f(to, xo, >)