Answer
\begin{align*}
y(0)&=d \\
y'(0)&= c\\
y''(0)&= 2b\\
y'''(0)&= 6a\\
y^{(4)}(0)&= 24\\
y^{(5)}(0)&= 0
\end{align*}
Work Step by Step
Given $$y = x^4 + ax^3 + bx^2 + cx + d $$
We compute the derivatives
\begin{align*}
y'&= 4x^3+3ax^2+2bx+c\\
y''&= 12x^2+6ax+2b\\
y'''&= 24x+6a\\
y^{(4)}&= 24\\
y^{(5)}&= 0
\end{align*}
Now, we can plug in $x=0$
\begin{align*}
y(0)&=d \\
y'(0)&= c\\
y''(0)&= 2b\\
y'''(0)&= 6a\\
y^{(4)}(0)&= 24\\
y^{(5)}(0)&= 0
\end{align*}
