Key to "Derivatives of vector-valued functions Differentiate the following function...."

3kofbe

Answered question

2021-11-23

Derivatives of vector-valued functions Differentiate the following function.

r (t) = ⟨ t e^{- t}, t \ln t, t \cos t ⟩

Opeance1951

Beginner2021-11-24Added 26 answers

Step 1
Given,

r (t) = ⟨ t e^{- t}, t e n t, t \cos t ⟩

So, obtain differntiate individually

\frac{d}{d x} (t e^{- t}) = e^{- t} + t (e^{- t}) (- 1)

= (1 - t) e^{- t}

Step 2
Similarly,

\frac{d}{d x} (t \ln t) = \frac{t}{t} + 1 \times \ln t

= \ln t + 1

\frac{d}{d x} (t \cos t) = \cos t + t (- \sin t)

= \cos t - t \sin t

∴ r^{'} (t) = ⟨ (1 - t) e^{- t}, (1 + \ln t), (\cos t - t \sin t) ⟩

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