slope in a log-transformation

Given y(x) = A*exp(x+c). I'm converting the y-axis into a log-scale. Then I calculate the slope at a certain point. Am I'm allowed to re-transform this value dy(x)/dx? Hence: exp(dy(x)/dx)?

• Presuming $A$ to be positive, it seems odd to have a multiplicative constant term $A$ and an additive term in the $\exp$, since they serve the same function. $A\exp(x+c) = Ae^c \exp(x) = A^* \exp(x)$ (or if you prefer, $A\exp(x+c) = e^{x + c+\log(A)}=e^{x + c^*}$). – Glen_b Jan 11 '19 at 11:31
• Correct, didn't have these rules in my mind anymore. Thank you! – Ben Jan 11 '19 at 16:12

Exponentiating the derivative of the log is not the same as the derivative of the original function ($$\log$$ and $$\exp$$ aren't linear operators):
$$\frac{d}{dx} Ae^{x+c} = Ae^c \frac{d}{dx} e^x = Ae^{x+c}$$
If you take logs of $$Ae^{x+c}$$ you get $$\log A + c + x$$, and so the slope on the log scale is 1. If you exponentiate that you get $$e$$. Clearly that's not going to be correct; the slope isn't constant.