A question uniform convergence

Question

Let $X$ be a compact Hausdorff space, $a$ a continuous real-valued function on $X$, and for $t\in\mathbb{R}$ let $f_t(x)=\exp(ia(x))$ such that the function $t\mapsto f_t$ is continuous (where we use the sup norm on $C(X)$). Is it then true in general that $$\frac{f_t-1}{t}-ia$$ converges uniformly to $0$? if $X$ is a subset of the real numbers, one can probably use the mean value theorem or something, or a series expansion. But in general I am not sure how to prove it/whether it is correct.