Whether you can claim your child as a dependent, if your ex-spouse did not, is based on who the custodial parent is and whose qualifying child the child is determined to be for that year according to the IRS rules. The following information will help determine who can:

• take the child tax credit,
• use the qualifying child for head of household status, 
• take the credit for child and dependent care expenses,
• use the exclusion for dependent care benefits, and
• take the earned income credit.

Per IRS Publication 17 Your Federal Income Tax, page 27:

Children of divorced or separated parents (or parents who live apart). In most cases, because of the residency test, a child of divorced or separated parents is the qualifying child of the custodial parent. However, the child will be treated as the qualifying child of the noncustodial parent if all four of the following statements are true.

1. The parents:
   1. Are divorced or legally separated under a decree of divorce or separate maintenance,
   2. Are separated under a written separation agreement, or
   3. Lived apart at all times during the last 6 months of the year, whether or not they are or were married.
2. The child received over half of his or her support for the year from the parents.
3. The child is in the custody of one or both parents for more than half of the year.
4. Either of the following statements is true.
   1. The custodial parent signs a written declaration, discussed later, that he or she won't claim the child as a dependent for the year, and the noncustodial parent attaches this written declaration to his or her return. (If the decree or agreement went into effect after 1984 and before 2009, see Post-1984 and pre-2009 divorce decree or separation agreement, later. If the decree or agreement went into effect after 2008, see Post-2008 divorce decree or separation agreement, later.)  
   2. A pre-1985 decree of divorce or separate maintenance or written separation agreement that applies to 2019 states that the noncustodial parent can claim the child as a dependent, the decree or agreement wasn't changed after 1984 to say the noncustodial parent can’t claim the child as a dependent, and the noncustodial parent provides at least $600 for the child's support during the year.

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CAUTION: If a child is treated as the qualifying child of the noncustodial parent under the rules for children of divorced or separated parents (or parents who live apart) described earlier, see Applying the tiebreaker rules to divorced or separated parents (or parents who live apart), later.

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Applying the tiebreaker rules to divorced or separated parents or parents who live apart. If a child is treated as the qualifying child of the noncustodial parent under the rules described earlier for children of divorced or separated parents (or parents who live apart), only the noncustodial parent can claim the child as a dependent and the child tax credit or credit for other dependents for the child. However, only the custodial parent can claim the credit for child and dependent care expenses or the exclusion for dependent care benefits for the child, and only the custodial parent can treat the child as a dependent for the health coverage tax credit. Also, the noncustodial parent can't claim the child as a qualifying child for head of household filing status or the earned income credit. Instead, the custodial parent, if eligible, or other eligible person can claim the child as a qualifying child for those two benefits. If the child is the qualifying child of more than one person for these benefits, then the tiebreaker rules just explained determine whether the custodial parent or another eligible person can treat the child as a qualifying child.