I've been trying to get a good grip on Gauss's lemma and thought i'd do all the calculations explicitly on $S^2$ to understand what's going on. And so i've gotten that
$$Exp_p(v) = cos(||v||)p + sin(||v||)\frac{v}{||v||}$$.
And as such
$$f(w) = D_v Exp_p(w) = cos(||v||)\frac{<v,w>}{||v||^2}v + sin(||v||)\left(\frac{w}{||v||} - \frac{<v,w>}{||v||^2}v - \frac{<v,w>}{||v||}p\right)$$.
Here i get the problem that i seem to get
$$<f(u),f(w)> = cos^2(||v||)\frac{<v,w><v,u>}{||v||^2} + sin^2(||v||)\left( 2\frac{<v,w><v,u>}{||v||^2} + \frac{<w,u>}{||v||^2} - \frac{<v,w><v,u>}{||v||^3}\right)$$.
and i can't se why this should be $<u,w>$.
I'm probably overlooking something trivial so my question is what part of the calculation is wrong?