I presume that you are required here to use the "difference quotient" definition. That is that the derivative is defined as (Many people do not like using "dt" here, reserving that for after the limit). Yes, with , .
But then you have an error in your very next line. You have 20(t+ dt) equal to 20t+ dt. It should be 20t+ 20dt. So
And subtracting from that leaves . You have only "dt" rather than "20dt". Finally, dividing by "dt", and the limit of that as dt goes to 0 is 20+ 90t.
As your Q. stands, there is not a specific & stated requirement to use the "difference quotient" definition. Hence, the solution provided in post #2 is adequate. Although the solution given in post #3 is mathematically valid, it is based on an unjustified presumption and is akin to using a sledge hammer to crack a nut.
Al.