The lecturer is saying
we can then say that the gradient of this line is equal to the amount of that function that increases in this interval divided by the length of the interval
where the expression
the amount of that function that increases in this interval
is a little bit complex.
Is it grammatical and idiomatic and reasonable to simplify the expression as this
the increased amount of that function in this interval
Answer
It is probably just a slip of the tongue and he intended to say
we can then say that the gradient of this line is equal to the amount that the function increases in this interval divided by the length of the interval
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