The Envelope Theorem, Some Workings/Substitution

I have an application of the envelope theorem below, and would love some feedback on the steps taken to get from (2) – (3) and (3) – (4) and (4) – (5)

The overall question is: ‘derive the marginal value for human capital in the second year’ using the second-period value function, with parameters:

• human capital $k_2$
• savings $m_2$
• consumption good price $p_2$
• labour wage $w_2$
• labour supply $l_2$

The value function is given:

(1) $$V(k_2,m_2;p_2,w_2) = max_{c_2,l_2} u(c_2, l_2)$$
s.t $$ p_2c_2 = w_2k_2l_2+m_2$$

(2) With the budget constraint substituted into the objective we get:

$$V(k_2,m_2;p_2,w_2)=u(w_2,kl_2(k_2,m_2;p_2,w_2+m)/p_2,l_2(k_2,m_2;p_2,w_2)$$

(3) By the envelope theorem

$$frac{dV(k_2;m_2;p_2,w_2)}{dk}= [frac{d}{dk}u ((w_2kl_2+m_2)/p_2,l_2]_{l_2=l_2(k_2,m_2;p_2,w_2)}$$

(4) $$= [u_c((w_2kl_2+m_2/p_2,l_2)frac{w_2l_2}{p_2}]_{l_2=l_2=(k_2,m_2;p_2,w_2)}$$

(5) $$=u_c(k_2,m_2;p_2,w_2),l_2(k_2,m_2;p_2,w_2))frac {w_2l_2(k_2,m_2;p_2,w_2)}{p_2}$$

Screenshot uploaded for clarity