Answer:
dw/dt = (18t^9 - 9t^10)/(7+8t) - [t^9e^(2-t)/(7+8t)] - (16t^8)(2-t)*e^(2-t)/(7+8t)^2.
Explanation:
To find dw/dt, we can use the chain rule as follows:
dw/dt = (dw/dx)(dx/dt) + (dw/dy)(dy/dt) + (dw/dz)*(dz/dt)
First, let's find the partial derivatives of w with respect to x, y, and z:
dw/dx = ey/z
dw/dy = xe-y/z
dw/dz = -xey/z^2
Next, let's find dx/dt, dy/dt, and dz/dt:
dx/dt = 9
dy/dt = -1
dz/dt = 8t
Now, we can substitute these values into the chain rule formula:
dw/dt = (ey/z)(9) + (xe-y/z)(-1) + (-xey/z^2)*(8t)
Substituting x = t^9, y = 2-t, and z = 7+8t, we get:
dw/dt = [(2-t)t^9/(7+8t)]9 - [t^9e^(2-t)/(7+8t)] + [-t^9(2-t)*e^(2-t)/(7+8t)^2]*8t
Simplifying this expression, we get:
dw/dt = (18t^9 - 9t^10)/(7+8t) - [t^9e^(2-t)/(7+8t)] - (16t^8)(2-t)*e^(2-t)/(7+8t)^2
Therefore, dw/dt = (18t^9 - 9t^10)/(7+8t) - [t^9e^(2-t)/(7+8t)] - (16t^8)(2-t)*e^(2-t)/(7+8t)^2.