Contradiction in the derivative of the unit step function being the dirac delta function

Question

We know that $\delta(t)=\frac{d}{dt}u_c(t)$, where $u_c(t)$ is the unit step function. However, using this to find the Laplace transform of $\delta(t)$, $$\int_0^\infty{\delta(t)e^{-st}dt}=\int_0^\infty{\delta(t)dt}$$ $$=\int_0^\infty{\frac{d}{dt}u_c(t)dt}=u_c(\infty)-u_c(0)$$ But, $u_c(t)=1$ at $t\ge0$ so $u_c(\infty)-u_c(0)=1-1=0$. Isn't this contradictory to the Laplace transform of $\delta(t)$ being equal to $1$?