To use the chain rule to find dz/dt for the function z = xy3 + x2y + xt2y(t2 + 1), we must first identify all the dependent variables and how they relate to t. In this function, z depends on x and y, both of which are functions of t.
1. **Identify the Variables:**
We recognize that if x = f(t) and y = g(t), then z can be expressed as a function of t through these two variables.
2. **Differentiate using Chain Rule:**
According to the chain rule, the derivative of z with respect to t is given by:
dz/dt = (dz/dx)*(dx/dt) + (dz/dy)*(dy/dt)
3. **Find Partial Derivatives:**
We need to compute the partial derivatives:
dz/dx and dz/dy:
dz/dx = y3 + 2xy + (t2 + 1)y(t2 + 1)
dz/dy is also needed:
dz/dy = 3xy2 + x2 + x(t2 + 1)
4. **Plugging Back:**
Now plug dz/dx and dz/dy into the chain rule equation:
dz/dt = (y3 + 2xy + (t2 + 1)y(t2 + 1))*(dx/dt) + (3xy2 + x2 + x(t2 + 1))*(dy/dt)
5. **Conclusion:**
Once the derivates dx/dt and dy/dt are known, you can substitute them back into the equation to get the final result.