If $$ f(t)=e^{-\sin t+4 \cos t} $$, find $$ f^{\prime}(t) $$. Use exact values. $$ f^{\prime}(t)=\square $$

Question

UpStudy AI · Accepted Answer

We start with the function

$$
f(t)=e^{-\sin t+4\cos t}.
$$

Let

$$
g(t)=-\sin t+4\cos t,
$$

so that

$$
f(t)=e^{g(t)}.
$$

The derivative of $$ f(t) $$ with respect to $$ t $$ is given by

$$
f'(t)=e^{g(t)}\cdot g'(t).
$$

We next find the derivative $$ g'(t) $$. Differentiating $$ g(t) $$ term by term, we have

$$
\frac{d}{dt}(-\sin t) = -\cos t\quad \text{and}\quad \frac{d}{dt}(4\cos t) = -4\sin t.
$$

Thus,

$$
g'(t)=-\cos t-4\sin t.
$$

Substituting back into the expression for $$ f'(t) $$, we obtain

$$
f'(t)=e^{-\sin t+4\cos t}\cdot(-\cos t-4\sin t).
$$

This result can also be written as

$$
f'(t)=-\left(\cos t+4\sin t\right)e^{-\sin t+4\cos t}.
$$

So the exact value is:

$$
f'(t)= -\left(\cos t+4\sin t\right)e^{-\sin t+4\cos t}.
$$
$$ f^{\prime}(t) = -(\cos t + 4\sin t)e^{-\sin t + 4\cos t} $$

If \( f(t)=e^{-\sin t+4 \cos t} \), find \( f^{\prime}(t) \). Use exact values. \( f^{\prime}(t)=\square \)