Final answer:
To find dz/dt using the chain rule, we differentiate x and y with respect to t, apply the chain rule formula dz/dt = dz/dx * dx/dt + dz/dy * dy/dt, and substitute the values to find dz/dt.
Step-by-step explanation:
To find dz/dt using the chain rule, we will start by finding the derivatives of x and y with respect to t. Given x = 19t, the derivative of x with respect to t is dx/dt = 19. Given y = 1 - t^19, the derivative of y with respect to t is dy/dt = -19t^18. Now, we can apply the chain rule to find dz/dt.
Using the chain rule, we have dz/dt = dz/dx * dx/dt + dz/dy * dy/dt. We already know that z = (x + y)e^y, so dz/dx = e^y and dz/dy = (x + y)e^y + e^y * dy/dt.
Substituting the values we found earlier, dz/dx = e^(1 - t^19) and dz/dy = (19t + 1)e^(1 - t^19) - 19t^18 * e^(1 - t^19). Finally, substituting dz/dx, dx/dt, dz/dy, and dy/dt into the chain rule formula, we can calculate dz/dt as dz/dt = e^(1 - t^19) * 19 + (19t + 1)e^(1 - t^19) - 19t^18 * e^(1 - t^19).