Please Help me to fill in the gaps

Show 

$$ \frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$

---

$$\frac{\text d \langle {p} \rangle}{ \text{d} t} $$ 
$$= \frac{\text d}{\text d t} \int\limits\_{-\infty}^{\infty} \Psi^\* \left( \frac{\hbar}{i}\frac{\partial}{\partial x} \right) \Psi ~\text d x$$
$$= \frac{\hbar}{i}\int\frac{\partial }{\partial t} \left(\Psi^\*\frac{\partial\Psi}{\partial x}\right)~\text d x $$
$$= \frac{\hbar}{i}\int \frac{\partial \Psi^\*}{\partial t}\frac{\partial \Psi}{\partial x} +\Psi^\*\frac{\partial }{\partial x}\frac{\partial \Psi }{ \partial t} ~\text {d} x$$
$$= \frac{\hbar}{i}\int\left( -\frac{i\hbar}{2m}\frac{\partial^2 \Psi^\*}{\partial x^2}+\frac{i}{\hbar} V\Psi^\*\right)\frac{\partial \Psi}{\partial x}+\Psi^\* \frac{\partial}{\partial x} \left(\frac{i\hbar}{2m} \frac{\partial^2 \Psi }{\partial x^2} -\frac{i}{\hbar}V\Psi \right)~\text d x$$
$$=\int\left(V\Psi^\*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^\*}{\partial x^2}\right) \frac{\partial \Psi}{\partial x}+\Psi^\*\frac{\partial}{\partial x}\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}-V\Psi\right)\text d x$$

$$=\left. \left(V\Psi^\*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^\*}{ \partial x^2} \right) \Psi \right|\_{-\infty}^{\infty}-\int\left(\frac{\partial}{\partial x} (V\Psi^\*)-\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^\*}{ \partial x^3}\right)\Psi \text d x+ \qquad\qquad\left.\Psi^\*\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} - V\Psi\right)\right|\_{-\infty}^\infty-\int\frac{\partial\Psi^\*}{\partial x} \left( \frac{\hbar^2}{2m}\frac{\partial ^2 \Psi}{\partial x^2} - V \Psi \right)\text d x$$

$$=0 + \int\left(\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^\*}{ \partial x^3}-\frac{\partial}{\partial x} (V\Psi^\*)\right) \Psi \text d x+0+\int\frac{\partial\Psi^\*}{\partial x}\left(V\Psi - \frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}\right) \text d x$$
$$=\int\frac{\hbar^2}{2m} \left(\frac{\partial^3\Psi^\*}{\partial x^3}\Psi -\frac{\partial\Psi^\*}{\partial x} \frac{\partial^2\Psi}{\partial x^2}\right)+\frac{\partial\Psi^\*}{\partial x}(V\Psi )-\frac{\partial}{\partial x}(V\Psi^\*)\Psi\text d x $$

$$\vdots$$
$$=\int \frac{\hbar^2}{2m}\left( \Psi^\* \frac{\partial^3 \Psi}{\partial x^3} -\frac {\partial^2 \Psi^\*}{\partial x^2} \frac{\partial \Psi}{\partial x} \right)+\left( V \Psi^\* \frac{\partial \Psi}{\partial x}-\Psi^\* \frac{\partial}{\partial x} (V \Psi) \right)~\text d x$$
$$\vdots $$
$$= \int \left( V \Psi^\* \frac{\partial \Psi}{\partial x}-\Psi^\*V \frac{\partial \Psi}{\partial x} - \Psi^\* \frac{\partial V}{\partial x} \Psi \right)~\text d x$$
$$= \int\limits\_{-\infty}^{\infty} -\Psi^\* \frac{\partial V}{\partial x} \Psi ~\text {d} x$$
$$=\left\langle - \frac{ \partial V }{\partial x} \right\rangle $$

---

**Edit:**

Problem 1.7 (Introduction to Quantum Mechanics, 2edJ -David G. Griffiths)

Calculate $ \frac{\text d \langle {p} \rangle}{ \text{d} t}$

The solutions manual for the text provides this incomplete method, I haven't worked out all the details

![solution in solutions manuel](https://i.stack.imgur.com/4l9Sb.png)