How does Gradient Descent treat multiple features? As far as I know, when you reach the step, in a gradient descent algorithm, to calculate step_size, you calculate learning_rate * slope
Now, slope is obtained by calculating the derivative of the cost_function with respect to the feature you want to find the optimal coefficient for.
Let's say that the cost function for the purposes of this question is the sum of squared residuals.
My question is, how are coefficients of other features treated in the differentiation of the equation? For instance, if I have the equation $y = b_0 + x_1 + x_2$, then by calculating the derivative of the cost function with respect to $x_1$, one gets:
$\frac{d}{d\left(x_1\right)}\left(\left(\hat{y}\:-\:\left(b_0\:+\:x_1+x_2\right)\right)^2\right) =$
$2\times\:\left(\hat{y}\:-\:b_{0\:}-x_1-x_2\right)\left(\frac{d}{d\left(x_1\right)}(x_2)\:+\:1\right)$
In this case, how is a value obtained by substituting a value for $x_1$ while $\frac{d}{d(x_1)}(x_2)$ is still in the formula?
I watched a YouTube video (it starts at the right point) that says that  $x_2$ is a constant (while it's a different feature) and, therefore, when differentiating, $\frac{d}{d(x_1)}(x_2)$ is omitted and we are left with $2\times \:\left(\hat{y}\:-\:b_{0\:}-x_1-x_2\right)(1)$. Is this the case or am I missing something?
That's correct.  The derivative of $x_2$ with respect to $x_1$ is 0.
A little context: with words like derivative and slope, you are describing how gradient descent works in one dimension (with only one feature / one value to optimize).  In multiple dimensions (multiple features / multiple variables you are trying to optimize), we use the gradient and update all of the variables in each step.  That said, yes, this is basically equivalent to separately updating each variable in the one-dimensional way that you describe.