Answer
${\dfrac{dP}{dt}}={2RI{\dfrac{dI}{dt}}+I^2{\dfrac{dR}{dt}}}$;
and ${{\dfrac{dR}{dt}}=-\dfrac{2P}{I^3}{\dfrac{dI}{dt}}}=-\dfrac{2R}{I}{\dfrac{dI}{dt}}$
Work Step by Step
Since, $P=I^2R$
Now, on differentiating each side with respect to time, we get
${\dfrac{dP}{dt}={R{\dfrac{dI^2}{dt}}+I^2{\dfrac{dR}{dt}}}}$
or, ${\dfrac{dP}{dt}}={2RI{\dfrac{dI}{dt}}+I^2{\dfrac{dR}{dt}}}$
or, $0={2RI{\dfrac{dI}{dt}}+I^2{\dfrac{dR}{dt}}}$
or, ${2RI{\dfrac{dI}{dt}}=-I^2{\dfrac{dR}{dt}}}$
or, ${{\dfrac{dR}{dt}}=-\dfrac{2RI}{I^2}{\frac{dI}{dt}}}$
or, ${{\dfrac{dR}{dt}}=-\dfrac{2P}{I^3}{\dfrac{dI}{dt}}}$
Thus,
${\dfrac{dP}{dt}}={2RI{\dfrac{dI}{dt}}+I^2{\dfrac{dR}{dt}}}$
and ${{\dfrac{dR}{dt}}=-\dfrac{2P}{I^3}{\dfrac{dI}{dt}}}=-\dfrac{2R}{I}{\dfrac{dI}{dt}}$
