How do I compute the fundamental group of a disk?

I need to show that $\Pi_1(D^2,x)=\{id\}$ for all $x$ in the Disk. I somehow don't see how I can compute the fundamental group, since we only have the following definition:

The fundamental group of $X$ with base point $x$ is $$\Pi_1(X,x)=\{\text{continuous loops based ad x}\}/\{\text{homotopies that fixes base points}\}$$ (thats really our definition}

Since this topic is really new, I don't see where to start with such an exercise. Could someone take some time to discuss it with me, I don't want you to give me a full solution since I don't think that this would help alot.

Thank you

The fundamental group $\pi_1(X)$ of a topological space $X$ is defined as the group of continuous closed curves (loops) up to homotopy, that is, given the set of all continuous closed curves in $X$, we identify those curves that are homotopic to each other (this set carries a group structure). Two curves $f,g\colon I\to X$ are homotopic if there exists some homotopy $H\colon I\times I \to X$ so that $$H(0,\cdot) = f,\quad H(1,\cdot) = g$$

Now, given a convex set $K$, you can always build the so called straight line homotopy, that is, given two paths $g,f\colon I \to K$ with the same starting- and endpoints, there exists the homotopy

\begin{align} H\colon K\times I &\to K \\ (x,t) &\mapsto f(x) + t(g(x) - f(x)) \\ &= f(x)(1-t) + tg(x) \end{align}

Clearly, $D^2\subset \mathbb R^2$ is convex, thus, any arbitrary closed curve will be homotopic to the constant loop at the origin.

Now if all loops in your favorite space $X$ are homotopic to the constant loop (which is the identity element in $\pi_1(X)$), then $\pi_1(X)$ is simply the trivial group, i.e., consists up to homotopy only of the trivial element, i.e. the constant loop.

Addendum: Observe that for path-connected spaces, $\pi(X,x_0)$ is independed of the choice of basepoint $x_0$.

Hint. Fix $x_0$. Note that given any point $y$ in the disc, the line joining $x_0$ and $y$ is also completely within the disc. Using this, try to figure out a homotopy between an arbitrary loop and the constant loop at $x_0$. Think of functions like $$(s, t) \mapsto (1 - t)\sigma(s) + tx_0.$$