Final answer:
To compute dw/dt for w = xey/z with x = t³, y = 8-t, and z = 5t, we use the chain rule to differentiate w with respect to x, y, and z, then multiply these by the derivatives of x, y, and z with respect to t, and substitute the functions back.
Step-by-step explanation:
To find dw/dt using the chain rule, we need to differentiate w with respect to t while considering that w is a function of x, y, and z, which in turn are each functions of t. Since w = xey/z, we can express the total derivative dw/dt as:
dw/dt = (∂w/∂x)(dx/dt) + (∂w/∂y)(dy/dt) + (∂w/∂z)(dz/dt)
Substituting the given functions:
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- x(t) = t³
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- y(t) = 8 - t
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- z(t) = 5·t
we first find the partial derivatives of w:
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- ∂w/∂x = ey/z
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- ∂w/∂y = xe/z
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- ∂w/∂z = -xey/z²
and the derivatives of x, y, and z with respect to t:
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- dx/dt = 3t²
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- dy/dt = -1
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- dz/dt = 5
Now we multiply each partial derivative of w by the corresponding derivative of x, y, and z with respect to t:
dw/dt = (ey/z)(3t²) + (xe/z)(-1) + (-xey/z²)(5)
Finally, we substitute x, y, and z back into the partial derivatives:
dw/dt = ((8 - t)e^(t³)/5t)(3t²) + (t³e^(t³)/(5·t))(-1) + (-(t³)(8 - t)e^(t³)/(5·t)²)(5)
This represents the rate of change of w with respect to t.