Step 1 :Given $f(x)=-10|x-4|+1$, we are asked to find $f^{\prime}(4)$.
Step 2 :The derivative of a function at a point gives us the slope of the tangent line at that point.
Step 3 :However, for absolute value functions, the derivative at the point where the function changes direction (in this case, x=4) does not exist because the function is not differentiable at that point.
Step 4 :The slope of the tangent line is not defined because the function has a sharp turn at that point.
Step 5 :Therefore, the derivative of the function f(x) at x=4 does not exist.
Step 6 :Final Answer: \(\boxed{\text{Does not exist}}\)