This problem is from Differential Equations with Applications and Historical Notes by George F. Simmons. 3rd edition. 1.4. problem 10
This problem is about bacteria growth.
Living space for the colony of bacteria is limited and food is supplied at a constant rate, so that competition for food and space acts in such a way that ultimately the population will stabilize at a constant level $x_1$ ($x_1$ can be thought of as the largest population sustainable by this environment). Assume further that under these conditions the population grows at a rate proportional to the product of $x$ and the difference $x_1 − x$, and find x as a function of t. (initial amount of bacteria is $x_0$.)
$dx/dt=kx(x_1-x)$ where k is constant. when integrated,
$x=\frac{x_0x_1}{x_0+(x_1-x_0)e^-kx_1t}$
I agree with the solution.(This is the solution of the textbook too.)
But my question is, when I differentiate above solution back to $dx/dt$,
$dx/dt=\frac{kx_0x_1^2(x_1-x_0)e^-kx_1t}{(x_0+(x_1-x_0)e^-kx_1t)^2}$
And
$dx/dt=kx(x_1-x)\neq dx/dt=\frac{kx_0x_1^2(x_1-x_0)e^-kx_1t}{(x_0+(x_1-x_0)e^-kx_1t)^2}$
I can't understand the meaning of this. Aren't they supposed to be same? Clearly I am missing something.