**Partial Derivatives****Overview:**We consider the function \( P(T, F) = e^{\sqrt{F}(1 + 4T)^{3/2}} \), which models the population of cells \( P \) as it depends on two key factors: the ambient temperature \( T \) (in degrees Celsius) and the availability of a liquid "food" \( F \) (in mL).**Tasks:**(a) Calculate \( P_T(2,4) \) and interpret its meaning, including proper units.(b) Calculate \( P_F(2,4) \) and interpret its meaning, including proper units.(c) Calculate \( P_{FF}(2,4) \) and interpret its meaning, including proper units.(d) Calculate \( P_{FT}(2,4) \) and interpret its meaning, including proper units.These tasks involve finding the partial derivatives of the function \( P(T, F) \). Partial derivatives help us understand how the population \( P \) changes with respect to either the ambient temperature \( T \) or the availability of liquid "food" \( F \), while keeping the other variable constant. Each derivative computation and its interpretation will provide insights into the sensitivity of the cell population to changes in temperature and food availability.

Given: PT ,F=eF1+4T32 To Find: a PT2 ,4b PF2 ,4c PFF2 , 4d PFT2 ,4(a) PT ,F=eF1+4T32 Partially differentiate P(T ,F) with respect to T, treating F as constant ∂P∂T=eF321+4T12×4∂PT ,F∂T=6eF1+4T12∂P∂T=PT=6eF1+4T12PT2 ,4=6e41+4212 PT2 ,4=6e2912PT2 ,4=6e2×3PT2 ,4=18e2 CmLPT2 , 4=133.003 Cml The rate of change of population with respect to temperature is 199.504 C mL.b PT ,F=eF1+4T32 ∂∂FPT ,F=eF×12F1+4T32 ∂∂FPT ,F=eF2F1+4T32 ∂∂FP2 ,4=e4241+4232 ∂∂FP2 ,4=e24932 ∂∂FP2 ,4=e24×27∂∂FP2 ,4=27e24∂∂FP2 ,4=49.876128 CmL The rate of change of population with respect to food is 49.876128C mL. (c) PT ,F=eF1+4T32 ∂∂FPT ,F=eF2F1+4T32 ∂∂FPT ,F=eF2F1+4T32∂2∂F2PT , F=1+4T322FeF×12F-eF ×12FF2∂2∂F2PT , F=1+4T322eF2-eF2FF2∂2∂F2PT , F=1+4T324FeF-eFFFPFF2 ,4=1+83244e4-e444PFF2 ,4=2742e2-e24×2PFF2 ,4=2732e2CmLPFF2 ,4=6.23451 CmL PFF2 , 4 is giving the slope of the tangent line at the point (2, 4).d PT ,F=eF1+4T32 ∂2∂T∂FPT ,F=eF2F321+4T12×4∂2∂T∂FPT ,F=eF2F61+4T12∂2∂T∂FPT ,F=3eFF1+4T12PFTT, F=3eFF1+4T12PFT2, 4=3e441+4212PFT2, 4=9e22CmLPFT2, 4=33.25075 CmL PFT2, 4 is twist.