a few general considerations;
Know what you need - never mind what the company's expectation is - know your own bottom line. Be willing to walk away if you don't see that bottom line met.
Know your BATNA (Best Alternative to a Negotiated Agreement). This means - if you DON'T take this gig - what's your next best thing? Your BATNA helps you weigh their offer against what else you may choose instead, helps you assess the relative value of any offer and helps you stay 'rooted' in confidence that you can walk away if you need to.
Know THEIR BATNA - negotiations are about compromise and arriving at mutually-beneficial outcomes. You need to have in mind what the employer's perspective is. If they don't hire you - what will they miss out on? What is their alternative? What is their likely willingness to choose that alternative over you.
Have a general understanding of your market value - use online references if you don't have a clear understanding of that from primary sources (like other job opportunities).
Know your OPTIONS - in addition to salary, many employers have other benefits that may come into play. You mentioned child care as an issue for me. Often, employers will have a child care benefit option. If you're unable to negotiate a salary above your strike point - then bring up 'other benefits'. "I think we have a great fit here and I'd love to be a part of your team. However, at the salary we're discussing, I'll be challenged to do my best for you. Are there any creative options we can discuss around child care allowance?". Sometimes companies have company-sponsored programs or can offer an allowance that includes tax breaks for the company..... so, go into the negotiation prepared to discuss non-cash benefits that offset your expenses.
Be prepared, be confident, be relaxed, be matter-of-fact, be enthusiastic and focused on the VALUE you bring and the VALUE you expect in return. And be willing to walk away respectfully if need be..... "I really wish we could find an agreeable outcome here. I believe I'd be a very strong asset to your team and would enjoy working here - unfortunately we just aren't in the right realm of value with these discussions. If something should change, please feel free to reach back out."
Good luck