1. **Stating the problem:** We have a computational graph with variables $x$, $p = \cos(x)$, $y$, and $L = p \cdot y^2$. The inputs $x$ and $y$ depend on time $t$ as $x(t) = t^2$ and $y(t) = e^{3t}$. We need to: (a) Draw the tree diagram showing dependencies from $t$ to $L$. (b) Calculate the total derivative $\frac{dL}{dt}$ at $t=0$. 2. **Tree Diagram Explanation:** - $t$ influences $x$ and $y$. - $x$ influences $p$ via $p = \cos(x)$. - $p$ and $y$ influence $L$ via $L = p \cdot y^2$. 3. **Calculating $\frac{dL}{dt}$:** Using the chain rule: $$ \frac{dL}{dt} = \frac{\partial L}{\partial p} \frac{dp}{dt} + \frac{\partial L}{\partial y} \frac{dy}{dt} $$ Calculate each partial derivative: $$ \frac{\partial L}{\partial p} = y^2 $$ $$ \frac{\partial L}{\partial y} = 2 p y $$ Calculate derivatives of $p$ and $y$ with respect to $t$: $$ \frac{dp}{dt} = \frac{dp}{dx} \frac{dx}{dt} = -\sin(x) \cdot \frac{d}{dt}(t^2) = -\sin(x) \cdot 2t $$ $$ \frac{dy}{dt} = \frac{d}{dt} e^{3t} = 3 e^{3t} $$ 4. **Evaluate all at $t=0$:** $$ x(0) = 0^2 = 0 $$ $$ y(0) = e^{0} = 1 $$ $$ p(0) = \cos(0) = 1 $$ $$ \frac{dp}{dt}\bigg|_{t=0} = -\sin(0) \cdot 2 \cdot 0 = 0 $$ $$ \frac{dy}{dt}\bigg|_{t=0} = 3 e^{0} = 3 $$ 5. **Substitute values into total derivative:** $$ \frac{dL}{dt} = y^2 \cdot 0 + 2 p y \cdot 3 = 0 + 2 \cdot 1 \cdot 1 \cdot 3 = 6 $$ **Final answer:** $$ \frac{dL}{dt} \bigg|_{t=0} = 6 $$