Derivative of Survival Function

Question

I am trying to get through statistical survival analysis - sadly I only have high school math. I have the following equation:

$ S(t) = Pr\{T ≥ t\} = 1−F(t) = \int_t^\infty f(x) dx$

$f(x)$ is the probability density function. $F(t)$ is the cumulative distribution function, $S(t)$ is the survival function.

Apparently the derivative of $S(t)$ is $-f(t)$. I can't work out how to get that from the function above. Does the fact that $ F(t) = \int_0^t f(x) dx$ have anything to do with it? Or can it be worked out from the last equation?

Thanks for helping,

Steph

Answer 1

Just take the derivative of what you have before your last equality: $$ \frac{d S(t)}{d t}=\frac{d}{dt}(1-F(t))=-\frac{d}{dt}F(t)=-f(t). $$

Answer 2

$$ S(t) = 1−F(t) = 1-\int_0^t f(x)dx$$

$$\frac{dS(t)}{dt}=0-\frac{d}{dt}\int_0^t f(x)dx$$

The last part by the Fundamental Theorem of Calculus is exactly $f(t)$.

$$\frac{d}{dt}\int_0^t f(x)dx=f(t)$$