Please tell me whether this sentence is correct.
The first step to correcting a mistake is patience.
I think that after to the verb must be in its infinitive form. Am I mistaken?
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There are actually two different tos.
One is the infinitive marker, which tells us that what follows is the infinitive form of the verb.
I want to correct my mistakes.
The other is the preposition, which marks its object—the noun or noun phrase which follows it—as the direction or goal of an action.
He is driving to London.
In your example, the to is in fact the preposition: you take a first step to or toward a goal. (In fact, most native speakers would use toward rather than to). Consequently it must take the form of the verb which can act as a noun: the -ing form, which is called a gerund.
The first step toward correcting a mistake is patience.
(Note that the marked infinitive form of the verb can be used as a noun, too—for instance, To correct one’s mistakes is virtuous—but it is not used as the object of a preposition.)
After "to" a verb must be in its infinitive form. You are not mistaken there. Then it must be:
The first step to correct a mistake is patience.
But "correcting" can also be a substantive, i.e. it is also possible:
The first step to (the) correcting of a mistake is patience.
In that case the of would be missing (while the correcting would be right in this case).
It is also possible to use for plus gerund, i.e.:
The first step for correcting a mistake is patience.
Here correcting is right, but to is wrong and needs to be replaced by for.