For the expression lnx2 y2 x7 8 calculate the derivative

Question

a) For the expression ln(x² + y²) = x⁷ + 8, calculate the derivative with respect to x.

b) The Palermo Technical Impact Hazard Scale P is used to quantify the risk associated with the impact of an asteroid colliding with the earth: P = log10 [ ( p_i E^0.8) / (0.03T) ]

where p_i is the probability of impact, T is the number of years until impact, and E is the energy of impact (in megatons of TNT). The risk is greater than a random event of similar magnitude if P > 0. Calculate (dP/dT) assuming that pi = 2 x 10^-5 and E = 9 megatons.