Why is MCMC needed when estimating a parameter using MAP Given the formula for MAP estimation of a parameter

Why is a MCMC (or similar) approach needed, couldn't I just take the derivative, set it to zero and then solve for the parameter?
 A: If you know which family your posterior is from and if finding the derivative of that distribution is analytically feasible, that is correct. 
However, when you use MCMC, you are likely not going to be in that type of situation. MCMC is made for situations in which you have no clear analytical notion of how your posterior looks like.
A: MCMC is not needed (or suggested) in order to compute a MAP estimate. In general, you won't be able to solve for the root of the MAP objective's derivative. However, you can use numerical optimization to approximately maximize the objective. As noted, MCMC is a sampling method, whereas MAP estimation is an optimization problem. MAP estimation only requires us to be able to evaluate the (unnormalized) posterior distribution at a given $\theta$, and only gives us one piece of information about that posterior. By approximately sampling an intractable posterior, MCMC methods allow us to learn all information about the posterior distribution.
A: Most posteriors prove to be difficult to optimize analytically (i.e. by taking a gradient and setting it equal to zero), and you'll need to resort to some numerical optimization algorithm to do MAP.
As an aside: MCMC is unrelated to MAP.
MAP - for maximum a posteriori - refers to finding a local maximum of something proportional to a posterior density and using the corresponding parameter values as estimates.  It is defined as
$$
\hat{\theta}_{MAP} = \text{argmax}_{\theta} \, p(\theta \, | \, D)
$$
MCMC is typically used to approximate expectations over something proportional to a probability density.  In the case of a posterior, that's
$$
\hat{\theta}_{MCMC} = n^{-1} \sum_{i=1}^{n} \theta^{0}_{i} \approx \int_{\Theta}\theta \, p(\theta \, | \, D)d\theta
$$
where $\{\theta^{0}_{i}\}^{n}_{i=1}$ is a collection of parameter space positions visited by a suitable Markov chain.  In general, $\hat{\theta}_{MAP} \neq \hat{\theta}_{MCMC}$ in any meaningful sense.
The crux is that MAP involves optimization, while MCMC is based around sampling.
