Step 1: See what $[4x+3]$ can be.
The greatest integer $[4x+3]$ is always a whole number. Since $\cos^{-1}(u)$ only allows $-1\le u\le 1$, the only possible values are $-1,0,1$.
Step 2: Find the matching $x$ ranges.
$[4x+3]=-1$ gives $-1\le x<-\tfrac34$; $[4x+3]=0$ gives $-\tfrac34\le x<-\tfrac12$; $[4x+3]=1$ gives $-\tfrac12\le x<-\tfrac14$.
Step 3: Find $f$ on each piece.
$\cos^{-1}(-1)=\pi$, $\cos^{-1}(0)=\tfrac{\pi}{2}$, $\cos^{-1}(1)=0$. So $f$ is a flat (constant) value on each strip.
Step 4: Differentiate the flat pieces.
A constant has slope $0$, so $f$ is differentiable inside each open strip.
Step 5: Look at the jumps.
At $x=-\tfrac34$ and $x=-\tfrac12$ the value of $f$ jumps from one constant to another. A jump is not even continuous, so it cannot be differentiable there.
Step 6: Write the differentiable set.
So $f$ is differentiable on $[-1,-\tfrac14)$ except the two jump points:
\[ \left[-1,-\tfrac14\right)-\left\{-\tfrac34,-\tfrac12\right\}. \]
\[ \boxed{\left[-1,-\tfrac14\right)-\left\{-\tfrac34,-\tfrac12\right\}} \]