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Question

# If $F(x, y)=x^3-x y^2-y^4, x=2 \cos 3 t$, and $y=3 \sin t$, find dF/dt at t=0.

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Given $F\left(x,y \right)=x^{3}-xy^{2}-y^{4}$, $x=2\cos 3t$ and $y=3\sin t$

\begin{align*} \dfrac{dF}{dt}&=\dfrac{dF}{dx}\dfrac{dx}{dt}+\dfrac{dF}{dy}\dfrac{dy}{dt}\\ &=3x^{2}\cdot \left( -6\right)\sin 3t+\left(2xy-4y^{3} \right)\cdot \left(3\cos t\right)\\ &=-18x^{2}\sin 3t+3\cos t\left(12\cos 3t\sin t-12\sin t^{3} \right)\end{align*}

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