Help regarding property of unit impulse function
Integrals of the form

$$\int_{a}^bf(t)\delta^{(n)}(t-a)dt\tag{1}$$

with $\delta^{(n)}(t)$ being the $n^{th}$ generalized derivative of the Dirac delta impulse, are undefined. What is well-defined are integrals of the form

$$\int_{a^+}^bf(t)\delta^{(n)}(t-a)dt\tag{2}$$

or

$$\int_{a^-}^bf(t)\delta^{(n)}(t-a)dt\tag{3}$$

In $(2)$, the Dirac impulse (or its derivative) is outside the integration interval and, consequently, the integral evaluates to zero, whereas in $(3)$ $\delta^{(n)}(t-a)$ is inside the integration interval and the value of the integral is given by the first formula of the question.