I'm working through Fulton and Harris's Representation Theory, and I'm stuck on Exercise 8.27.
I'm trying to show that if $A$ is an algebra and $G$ is the Lie group of algebra automorphisms (interpreted as a subgroup of $GL(A)$) then the Lie algebra associated to $G$ is the algebra of derivations on $A$ ($Der(A)$).
I've shown that if $\gamma : [0,1] \to G$ is some path (writing $\gamma(t)$ as $\gamma_t$) with $\gamma_0 = id$ and $\gamma'_0 = X$ then just by differentiating with respect to $t$ $$ \gamma_t(ab) = \gamma_t(a)\gamma_t(b) $$ we get $$ \gamma'_t(ab) = \gamma'_t(a)\gamma_t(b) + \gamma_t(a)\gamma'_t(b), $$ which at $t=0$ gives $$ X(ab) = X(a)b + aX(b), $$ so $X$ is a derivation. But I can't show that any derivation must be a tangent to the identity, so that the associated Lie algebra to $G$ is actually $Der(A)$.
Any help with this would be great, thanks!