Plotting the Arrhenius curve without the rate constant

I was carrying out an experiment about finding the activation energy of the Maillard reaction using the Arrhenius equation. Since I didn't know the rate constant for the reaction due to the complex reaction mechanism of the Maillard Reaction, I tweaked the rate reaction to fit into the Arrhenius reaction.
$$\text{Rate}=k[\text{glucose}]^m[\text{asparagine}]^n \\\implies k=\frac{\text{Rate}}{[\text{glucose}]^m[\text{asparagine}]^n } \\\implies k=([\text{glucose}]^m[\text{asparagine}]^n)\,\frac{\mathrm d[\text{product}]}{\mathrm dt} \\\implies k=\frac{c}{t}$$ where $c=([\text{glucose}]^m[\text{asparagine}]^n)\,\mathrm d[\text{product}]$. By ln both sides, I got $\ln k=\ln c-\ln t$ which I then subbed into the Arrhenius equation to get $$\ln c-\ln t=-\frac{E\_\mathrm a}{R}\times\frac{1}{T}+\ln A$$ By manipulating, I got $$\ln t=\frac{E\_\mathrm a}{R}\times\frac{1}{T}-\ln A+\ln c$$ Which then produce the below graph based on the experimental data[![enter image description here](https://i.stack.imgur.com/Vq4uu.png)](https://i.stack.imgur.com/Vq4uu.png)
However, the gradient is positive which technically should be negative according to other Arrhenius plots I have seen. Does anyone know if this is correct or is there something I missed out? 


