To find the derivative dz/dt using the chain rule, we first need to identify the variables involved. Here, we have z as a function of x, y, and t. So the chain rule tells us that:
dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt) + (dz/dt)
We will calculate each of these partial derivatives step by step.
Step 1: Calculate Partial Derivatives
z = xy³ + x²y + xt² + y + t²dz/dx = y³ + 2xy + t²dz/dy = 3xy² + x + 1dz/dt = 2t
Step 2: Find dx/dt and dy/dt
Next, we need to find the derivatives dx/dt and dy/dt. These derivatives will depend on the relationship between x, y, and t. We need to know how x and y change with respect to t.
Step 3: Combine Using the Chain Rule
Once we have dx/dt and dy/dt, we can substitute everything back into our chain rule equation:
dz/dt = (y³ + 2xy + t²)(dx/dt) + (3xy² + x + 1)(dy/dt) + 2t
This final expression gives us dz/dt in terms of dx/dt, dy/dt, x, y, and t.