Let \( \mathbf{u}(t)=7 t^{3} \mathbf{i}+\left(t^{2}-7\right) \mathbf{j}-3 \mathbf{k} \) and \( \mathbf{v}(t)=e^{t} \mathbf{i}+3 e^{-t} \mathbf{j}-e^{3 t} \mathbf{k} \). Compute the derivative of the following function. \( \mathbf{u}(t) \cdot \mathbf{v}(t) \) Select the correct choice below and fill in the answer box(es) to complete your choice. A. The derivative is the vector-valued function B. The derivative is the scalar function

To compute the derivative of the function \( \mathbf{u}(t) \cdot \mathbf{v}(t) \), we first need to find the dot product of the two vector-valued functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \). 1. **Calculate the dot product \( \mathbf{u}(t) \cdot \mathbf{v}(t) \)**: \[ \mathbf{u}(t) = 7 t^{3} \mathbf{i} + (t^{2} - 7) \mathbf{j} - 3 \mathbf{k} \] \[ \mathbf{v}(t) = e^{t} \mathbf{i} + 3 e^{-t} \mathbf{j} - e^{3t} \mathbf{k} \] The dot product is calculated as follows: \[ \mathbf{u}(t) \cdot \mathbf{v}(t) = (7 t^{3})(e^{t}) + ((t^{2} - 7)(3 e^{-t})) + (-3)(-e^{3t}) \] Simplifying this gives: \[ \mathbf{u}(t) \cdot \mathbf{v}(t) = 7 t^{3} e^{t} + 3(t^{2} - 7)e^{-t} + 3 e^{3t} \] 2. **Now, compute the derivative of the dot product**: We will differentiate the expression \( 7 t^{3} e^{t} + 3(t^{2} - 7)e^{-t} + 3 e^{3t} \). Using the product rule and chain rule, we differentiate each term: - For \( 7 t^{3} e^{t} \): \[ \frac{d}{dt}(7 t^{3} e^{t}) = 7 \left(3 t^{2} e^{t} + t^{3} e^{t}\right) = 7 e^{t}(3 t^{2} + t^{3}) \] - For \( 3(t^{2} - 7)e^{-t} \): \[ \frac{d}{dt}(3(t^{2} - 7)e^{-t}) = 3 \left(2t e^{-t} - (t^{2} - 7)e^{-t}\right) = 3 e^{-t}(2t - (t^{2} - 7)) \] - For \( 3 e^{3t} \): \[ \frac{d}{dt}(3 e^{3t}) = 9 e^{3t} \] 3. **Combine the derivatives**: Now we combine all the derivatives: \[ \frac{d}{dt}(\mathbf{u}(t) \cdot \mathbf{v}(t)) = 7 e^{t}(3 t^{2} + t^{3}) + 3 e^{-t}(2t - (t^{2} - 7)) + 9 e^{3t} \] Thus, the derivative \( \frac{d}{dt}(\mathbf{u}(t) \cdot \mathbf{v}(t)) \) is a scalar function because the dot product of two vector functions results in a scalar. **Final Answer**: B. The derivative is the scalar function.