Lewis Mccarthy
01/12/2023 · Junior High School
Differentiate the following function. \( r(t)=\left\langle e^{t}, 7 e^{-t},-4 e^{4 t}\right\rangle \) \( r^{\prime}(t)=\langle\square, \square, \square \)
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Tutor-Verified Answer
Step-by-step Solution
To differentiate the function \( r(t) = \left\langle e^{t}, 7 e^{-t}, -4 e^{4t} \right\rangle \), we will differentiate each component with respect to \( t \).
1. The first component is \( e^{t} \):
\[
\frac{d}{dt}(e^{t}) = e^{t}
\]
2. The second component is \( 7 e^{-t} \):
\[
\frac{d}{dt}(7 e^{-t}) = 7 \cdot \frac{d}{dt}(e^{-t}) = 7 \cdot (-e^{-t}) = -7 e^{-t}
\]
3. The third component is \( -4 e^{4t} \):
\[
\frac{d}{dt}(-4 e^{4t}) = -4 \cdot \frac{d}{dt}(e^{4t}) = -4 \cdot (4 e^{4t}) = -16 e^{4t}
\]
Now, we can combine these results to write the derivative \( r'(t) \):
\[
r'(t) = \left\langle e^{t}, -7 e^{-t}, -16 e^{4t} \right\rangle
\]
Thus, the final answer is:
\[
r^{\prime}(t) = \langle e^{t}, -7 e^{-t}, -16 e^{4t} \rangle
\]
Quick Answer
\[
r^{\prime}(t) = \langle e^{t}, -7 e^{-t}, -16 e^{4t} \rangle
\]
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