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andrew11 [14]
3 years ago
13

Use the chain rule to find dw/dt. w = ln x2 + y2 + z2 , x = 7 sin(t), y = 9 cos(t), z = 6 tan(t)

Mathematics
1 answer:
DENIUS [597]3 years ago
3 0
By the chain rule,

\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{\partial w}{\partial x}\dfrac{\mathrm dx}{\mathrm dt}+\dfrac{\partial w}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}+\dfrac{\partial w}{\partial z}\dfrac{\mathrm dz}{\mathrm dt}

We have

\dfrac{\partial w}{\partial x}=\dfrac{2x}{x^2+y^2+z^2}
\dfrac{\partial w}{\partial y}=\dfrac{2y}{x^2+y^2+z^2}
\dfrac{\partial w}{\partial z}=\dfrac{2z}{x^2+y^2+z^2}

\dfrac{\mathrm dx}{\mathrm dt}=7\cos t
\dfrac{\mathrm dy}{\mathrm dt}=-9\sin t
\dfrac{\mathrm dz}{\mathrm dt}=6\sec^2t

So we have

\dfrac{\mathrm dw}{\mathrm dt}=\dfrac2{x^2+y^2+z^2}\left(7x\cos t-9y\sin t+6z\sec^2t\right)
\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{2\left(49\sin t\cos t-81\cos t\sin t+36\tan t\sec^2t\right)}{49\sin^2t+81\cos^2t+36\tan^2t}
\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{8\sin t(9-16\cos^4t)}{\cos t(36+13\cos^2t+32\cos^4t)}
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