Hints: Hazard rate and density function for the gamma distribution

Here you need to use the relationships between the survival function and the density function and hazard rate, as described in chapter 1 in ABG. For the differentiation you'll need to apply the chain rule, and use that \(\frac{d}{dx} \int_x^{\infty} g(u)du = -g(x)\). The incomplete gamma function \(\Gamma(k,\gamma t)\) can be calculated in R as "pgamma(gamma*t,k,lower=FALSE)*gamma(k)", where t is a vector of values for which to evaluate the incomplete gamma function.

Solution: To get a detailed solution of the problem, see the link at the bottom of this page.