Final answer:
To find dy/dx and dz/dx from the given parametric equations, calculate the derivatives of y and z with respect to t, then divide by the derivative of x with respect to t, providing the rates of change dy/dx and dz/dx.
Step-by-step explanation:
To find dy/dx and dz/dx given the parametric equations x = e^(-t) cos(8t), y = e^(-t) sin(8t), z = e^(-t), we need to find the derivatives of y and z with respect to t and then divide by the derivative of x with respect to t.
Firstly, the derivative of x with respect to t is:
dx/dt = -e^(-t) cos(8t) - 8e^(-t) sin(8t).
The derivative of y with respect to t is:
dy/dt = -e^(-t) sin(8t) + 8e^(-t) cos(8t).
The derivative of z with respect to t is:
dz/dt = -e^(-t).
Now, dy/dx can be found by dividing dy/dt by dx/dt:
dy/dx = (dy/dt) / (dx/dt).
Likewise, dz/dx is found by dividing dz/dt by dx/dt:
dz/dx = (dz/dt) / (dx/dt).
Substituting the previously found derivatives into these formulas gives us the desired results for dy/dx and dz/dx.