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expeople1 [14]
2 years ago
12

Use chain rule to find dw/dt w=ln(x^2 + y^2 + z^2)^(1/2) , x=sint, y=cost, z=tant

Mathematics
1 answer:
son4ous [18]2 years ago
4 0
The chain rule states that

\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}

Since \ln(x^2+y^2+z^2)^{1/2}=\dfrac12\ln(x^2+y^2+z^2), you have

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

and

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

Also, since x^2+y^2+z^2=\sin^2t+\cos^2t+\tan^2t=1+\tan^2t=\sec^2t, the derivative is

\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{\sin t\cos t}{\sec^2t}-\dfrac{\sin t\cos t}{\sec^2t}+\dfrac{\tan t\sec^2t}{\sec^2t}
\dfrac{\mathrm dw}{\mathrm dt}=\tan t
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Step2247 [10]

Answer:

p=10\sqrt[3]{54}x^2

Step-by-step explanation:

We are given parallelogram

Since, two opposite sides of any parallelogram are always equal

Let's assume first side =a

second side =b

so, we get

a=2\sqrt[3]{54}x^2

b=3\sqrt[3]{54}x^2

now, we can find perimeter

perimeter=2a+2b

so, we get

p=2(2\sqrt[3]{54}x^2)+2(3\sqrt[3]{54}x^2)

we can simplify it

p=4\sqrt[3]{54}x^2+6\sqrt[3]{54}x^2

p=10\sqrt[3]{54}x^2................Answer


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3 years ago
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Dexter was asked to mix the punch for his sister’s birthday party. He has to figure out how much water and pure concentrate to a
Zigmanuir [339]

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Since it says the ratio of the outside number to the inside number, take that number and see how many times it can fit into the outside number. In this case, it is 4, and than for the inside number, it is 3.

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