<s>[INST] Vector Functions of One Variable Question
A particle moves along the curve of the intersection of the cylinders $y=-x^2$ and $z=x^2$ in the direction in which $x$ increases. (All distances are in cm.) At the instant when the particle is at the point $(1,\,-1,\,1)$ its speed is $9$ cm/s, and the speed is increasing at a rate of $3$ cm/s$^2$. Find the velocity and acceleration of particle at that instant.
Solution (Well at least my attempt at one)
$\mathbf{r}(t) = x\mathbf{i} - x^2\mathbf{j} + x^2\mathbf{k}$
$\mathbf{v}(t) = \left(\mathbf{i} - 2x\mathbf{j} + 2x\mathbf{k}\right)\dfrac{dx}{dt}$
\begin{align*}
\implies ||\mathbf{v}|| &=\left(\sqrt{1^2 +4x^2+4x^2}\right) \left|\dfrac{d x}{dt}\right|\\
&=(\sqrt{1+8x^2})\dfrac{dx}{dt}
\end{align*}
$|\frac{dx}{dt}|=\frac{dx}{dt}$ since the particle is always increasing
$\implies$ at the point $(1,\,-1,\,1),\,\, ||\mathbf{v}|| = 9$, so $dx/dt = 3$ and the velocity at that point is 
$\mathbf{v}(t) = 3\mathbf{i} - 6\mathbf{j} + 6\mathbf{k}$
Now acceleration is where I have a bit of trouble 
$\mathbf{a}(t) =\dfrac{d^2x}{dt^2}(\mathbf{i} - 2x\mathbf{j} + 2x\mathbf{k}) + (\dfrac{dx}{dt})^2(- 2\mathbf{j} + 2\mathbf{k})$
But I don't exactly know what to do from here any help would be appreciated
 [/INST] As I mentioned in my comment above, you can figure out $\dfrac{d^2x}{dt^2}$ from your equation in $|\mathbf{v}|$, given that you know that the change of speed is 3cm/s/s.
\begin{eqnarray}
|\mathbf{v}| &=& (\sqrt{1+8x^2})\dfrac{dx}{dt}\\
\dfrac{d|\mathbf{v}|}{dt} &=& (\sqrt{1+8^2})\dfrac{d^2x}{dt^2} + \frac{8x}{\sqrt{1+8x^2}}\left(\dfrac{dx}{dt}\right)^2 = 3 \\
\dfrac{d^2x}{dt^2} &=& -7
\end{eqnarray}
Plugging this into your equation for $\mathbf{a}$ gives 
\begin{equation}\mathbf{a} = -7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}\end{equation}
and it can easily be checked that $|\mathbf{a}|=3$
</s>