Answer
$\dfrac{du}{dp}=2xy^3(1+6p)+3x^2y^2(pe^p+e^p)+4z^3(p\cos p+\sin p)$
Work Step by Step
Given: $u=x^2y^3+z^4$
Apply chain rule.
$\frac{du}{dp}=(2xy^3)\frac{dx}{dP}+(3x^2y^2)\frac{dy}{dP}+(4z^3)\frac{dz}{dp}$
Plug in the values of $x=p+3p^2,y=pe^p,z=p \sin p$ in $(2xy^3)\frac{dx}{dP}+(3x^2y^2)\frac{dy}{dP}+(4z^3)\frac{dz}{dP}$
$\dfrac{du}{dp}=2xy^3(1+6p)+3x^2y^2(pe^p+e^p)+4z^3(p\cos p+\sin p)$
Hence,
$\dfrac{du}{dp}=2xy^3(1+6p)+3x^2y^2(pe^p+e^p)+4z^3(p\cos p+\sin p)$
