Hi All,

I would be grateful for some advice / help on how to solve this Calculus - time factor problem. My Goal here - I am trying to find the TIME factor within the 1st derivative that will yield the same results as the original data shown below. I was able to find the derivative with D[ f, t] And tried to extract Coefficients from original function f[t] And then apply the coefficients to the form - a t^2+b t +c & then use the correct coefficients in this form within the derivative to find each value in the dataset at each time {1,2,3,4,etc} I am not sure this is even the correct approach.

Please could someone review & advise how I should do this.

Many thanks for your help & attention. Best regards, Lea...

Given the the following data set:-

data = {{1, 4.008668526800082`}, {2, 13.840674130266803`}, {3, 
        29.80944537853352`}, {4, 51.91498227160025`}, {5, 80.15728480946696`}, {6,
         114.53635299213369`}, {7, 155.0521868196004`}, {8, 
        201.70478629186715`}, {9, 254.49415140893387`}, {10, 
        313.4202821708005`}, {11, 378.48317857746724`}, {12, 
        449.682840628934`}, {13, 527.0192683252008`}, {14, 
        610.4924616662674`}, {15, 700.1024206521342`}, {16, 795.849145282801`}};

which yields the following function f(t) & plots line & data points :-

    f[time_] := 0.3134285681335234` + 0.6268571362666725` time + 3.` time^2

    functionf = 0.3134285681335234` + 0.6268571362666725` time + 3.` time^2;

 Show[ListPlot[data, PlotStyle -> Red], ListLinePlot[data]]

derivativeOfF = D[f[time], time]

CoefficientList[functionf /. {x -> a t^2 + b t + c}, t]

I don't know what to do after this point to find the accurate time factor. Many thanks for you help.

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