Suppose the pair of $x$ values for a sample is $x_0$, $x_1$.
The percentage change in $y$ for the $x_0$, $x_1$ pair is defined as:
$$ \Delta y\% =\frac{y(x_1)-y(x_0)}{y(x_0)}\cdot100 $$
Substituting the formula $y=c\times(1-e^{px})$ and simplifying results in: $$ \Delta y\%=\frac{e^{px_0}-e^{px_1}}{1-e^{px_0}} $$
Since you have both $x_0$ and $x_1$ for each value of $\Delta y\%$, you can use an optimisation routine to fit the formula above to your data, resulting in an estimate of $p$.
Reproducible example in python:
I use scipy.optimize.curve_fit to fit a noisy set of observations. You could remove the noise term, or amplify it, depending on the nature of your data.
Imports and data: I synthesise a dataset comprising pairs of $(x_0, x_1)$, and their corresponding $\Delta y (\%)$.
import numpy as np from matplotlib import pyplot as plt # # Data for testing # np.random.seed(20) p_true = -0.5 def calc_y_change_percent(X, p): x0, x1 = X.T y_at_x0 = (1 - np.exp(p * x0)) y_at_x1 = (1 - np.exp(p * x1)) y_change_pct = (y_at_x1 - y_at_x0) / y_at_x0 * 100 return y_change_pct n_samples = 100 x0 = np.random.uniform(0.1, 0.5, size=n_samples) x1 = x0 + np.random.uniform(0, 1, size=n_samples) X = np.column_stack([x0, x1]) y_change_pct = calc_y_change_percent(X, p_true) + np.random.randn(n_samples) / 5
Visualise data, estimate $p_{true}$, and report both $p_{true}$ and $\hat p$:
# # Fit # from scipy.optimize import curve_fit p_hat, p_cov = curve_fit(calc_y_change_percent, X, y_change_pct) print('Estimated p:', p_hat) # # Visualise # f, ax = plt.subplots(figsize=(5, 4), layout='tight') ax.set( xlabel='$x_0$', ylabel='$x_1$', title= '$x_0$ and $x_1$ pairs coloured by $\Delta y\%$' + '\n$p_{true}=$' + str(p_true) + '\n$\hat p=$' + str(p_hat.round(4).item()) ) m = ax.scatter(x0, x1, c=y_change_pct) f.colorbar(mappable=m, label='$\Delta y \%$')
