Using chain rule find $\large\frac{dw}{dt}$ for each of the following : $w=e^{xy}$ Where $x=t^{2},y=t^{3}$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

Toolbox:
• Chain rule: If $u= f(x,y)$ is differentiable and $x,y$ are functions of t then $\large\frac{du}{dt}=\frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y} \frac{dy}{dt}$
• The chain rule for $u(x,y,z)$ where $x,y,z$ are functions of t is similarly stated.
$w=e^{xy}$ where $x=t^2,y=t^3$
Step 1:
$\large\frac{dx}{dt}$$=2t,\large\frac{dy}{dt}$$=3t^2,\large\frac{\partial w}{\partial x}$$=ye^{xy},\large\frac{\partial w}{ \partial y}$$=xe^{xy}$
Step 2:
$\large\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y} \frac{dy}{dt}$
$\qquad=ye^{xy}.2t+xe^{xy}.3t^2$
$\qquad=e^{xy}.(2yt+3xt^2)$
$\qquad=e^{t^5}(2t^4+3t^4)$
$\qquad= 5t^4e^{t^5} \qquad(x=t^2,y=t^3)$

edited Aug 13, 2013 by meena.p