Given the space $C^n[0,1]$ of all real functions of class $C^n$ in $[0,1]$, let $\tilde{d}^j := d_\infty(f^{(j)},g^{(j)})$ a pseudometric $(j=1,\dots,n)$ on $C^n[0,1]$. Here $f^{(j)}$ mean the $j^{th}$ derivative of $f$, and $d_\infty$ is the uniform metric given by $d_\infty(f,g) = \max\{|f(x)-g(x)|: x \in[0,1]\}$.
Question: Characterize the quotient spaces $C^n[0,1]/\tilde{d}^j.$
Thanks in advance.
You are close, but are confusing the definition of your pseudometics: the elements of your spaces are equivalence classes of functions given by $\{y\in C^n:\tilde{d^j}(x,y)=0\}$. Again, $x$ and $y$ are functions in the preceding set. If it helps you remember/keep track, we could rewrite the classes as $\{g\in C^n:\tilde{d^j}(f,g)=0\}$, using variable letters more commonly associated with functions.
Now we plug in the definition of the $\tilde{d^j}$ seminorms to see that the elements of $C^n[0,1] / \tilde{d^j}$ are classes $\{g\in C^n:\max_{x\in[0,1]}|f^{(j)}(x)-g^{(j)}(x)|=0\}$. Notice that only now is $x$ a point.
Finally, we consider what this all means. If the maximum difference between two functions is zero, then the functions must be the same. So this is saying that our equivalence classes are classes of functions with the same $j^{th}$ derivative. And this means that they must differ by a function whose $j^{th}$ derivative is zero. These functions are just polynomials of degree less than $j$.
In conclusion, the elements of your space are classes $\{ f(x)+p_j(x):p_j(x) \text{ is a polynomial of degree less than } j\}$