If \( f(t)=e^{-\sin t+4 \cos t} \), find \( f^{\prime}(t) \). Use exact values. \( f^{\prime}(t)=\square \)
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To find \( f^{\prime}(t) \) for the function \( f(t) = e^{-\sin t + 4 \cos t} \), we will apply the chain rule. First, denote \( u(t) = -\sin t + 4 \cos t \). Then, we have: \[ f(t) = e^{u(t)} \] Using the chain rule: \[ f^{\prime}(t) = e^{u(t)} \cdot u^{\prime}(t) \] Next, we'll compute \( u^{\prime}(t) \): \[ u^{\prime}(t) = -\cos t - 4 \sin t \] Thus, substituting back into our derivative: \[ f^{\prime}(t) = e^{-\sin t + 4 \cos t} \cdot (-\cos t - 4 \sin t) \] So, the final expression for the derivative is: \[ f^{\prime}(t) = e^{-\sin t + 4 \cos t} (-\cos t - 4 \sin t) \] You can fill in the answer: \[ f^{\prime}(t) = e^{-\sin t + 4 \cos t} (-\cos t - 4 \sin t) \]