The image shows an exercise from a textbook related to using the chain rule in partial differentiation to find the total derivative $\frac{dz}{dt}$ at a given point.

Problem Breakdown:

1. Part (a): $z = x \cos(y), \quad \text{where} \quad x = \sin(t), \quad y = \cos(t)$ Find $\frac{dz}{dt}$.

2. Part (b): $z = x^2 + 2xy + y^2, \quad \text{where} \quad x = e^t \cos(t), \quad y = e^t \sin(t)$ Find $\frac{dz}{dt}$.

Solutions:

Part (a)

1. Expression for $z$: $z = x \cos(y)$

2. Given functions: $x = \sin(t), \quad y = \cos(t)$

   The total derivative $\frac{dz}{dt}$ using the chain rule is given by: $\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$

3. Partial derivatives: $\frac{\partial z}{\partial x} = \cos(y)$ $\frac{\partial z}{\partial y} = -x \sin(y)$

4. Derivatives of $x$ and $y$ with respect to $t$: $\frac{dx}{dt} = \cos(t)$ $\frac{dy}{dt} = -\sin(t)$

5. Substitute and simplify: $\frac{dz}{dt} = \cos(y) \cdot \cos(t) + \left(-x \sin(y)\right) \cdot (-\sin(t))$ $\frac{dz}{dt} = \cos(\cos(t)) \cdot \cos(t) + \sin(t) \cdot \sin(\cos(t)) \cdot \sin(t)$ $\frac{dz}{dt} = \cos(\cos(t)) \cdot \cos(t) + \sin^2(t) \cdot \sin(\cos(t))$

Part (b)

1. Expression for $z$: $z = x^2 + 2xy + y^2$

2. Given functions: $x = e^t \cos(t), \quad y = e^t \sin(t)$

   Again, use the chain rule: $\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$

3. Partial derivatives: $\frac{\partial z}{\partial x} = 2x + 2y$ $\frac{\partial z}{\partial y} = 2x + 2y$

4. Derivatives of $x$ and $y$ with respect to $t$: $\frac{dx}{dt} = e^t \cos(t) - e^t \sin(t)$ $\frac{dy}{dt} = e^t \sin(t) + e^t \cos(t)$

5. Substitute and simplify: $\frac{dz}{dt} = (2x + 2y) \cdot \frac{dx}{dt} + (2x + 2y) \cdot \frac{dy}{dt}$ Simplify further by substituting the values of $x$ and $y$ and their derivatives.

Would you like further steps for the simplification, or do you have any specific questions?

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Follow-up Questions:

1. How does the chain rule apply when dealing with multiple variables?
2. Can you derive the partial derivatives of a more complex function?
3. What are the key differences between total and partial derivatives?
4. How would you handle a similar problem with more variables?
5. What happens if one of the functions is implicitly defined?

Tip: Always keep track of the variables being differentiated to avoid mistakes, especially when multiple variables depend on the same parameter.