Answer
$\dfrac{dR}{dt}=0.257023221$
The unit of the derivative is joules per square meter per second per kelvin.
Work Step by Step
The Stefan–Boltzmann Law gives $R = σT^4$.
To find $\dfrac{dR}{dt}$ use chain rule.
That is, $\dfrac{dR}{dt}=\dfrac{dR}{dT}\times\dfrac{dT}{dt}$
Now, to find $\dfrac{dR}{dT}$.
Derivate $R = σT^4$ with respect to temperature $T$.
We get, $\dfrac{dR}{dT}=\dfrac{d}{dT}(σT^4)$
Now, use the power rule to solve further.
$\dfrac{dR}{dT}=4σT^3$
Now substitute $\dfrac{dR}{dT}=4σT^3$ and $\dfrac{dT}{dt}= 0.05 K/year$ in $\dfrac{dR}{dt}=\dfrac{dR}{dT}\times\dfrac{dT}{dt}$.
$\dfrac{dR}{dt}=4\sigma T^{3}\times 0.05$
Substitute $\sigma = 5.67 ×10^{−8} Js^{−1}m^{−2}K^{−4}$ and $T = 283$
We get, $\dfrac{dR}{dt}=4\times5.67 ×10^{−8}\times283^{3}\times0.05=0.257023221$
Since the unit of $R$ is in joules per square meter per second and the unit of $T$ is in kelvins.
The unit of the derivative is the quotient of the units of $R$ and $T$.
That is, joules per square meter per second per kelvin.
