Solved: "Differentiate the vector-valued function r(t)=⟨sin⁡(t2),4t2−t⟩."

Coroware

Answered question

2021-11-30

Differentiate the vector-valued function

r (t) = ⟨ \sin (t^{2}), 4 t^{2} - t ⟩

John Timms

Beginner2021-12-01Added 8 answers

Step 1
Given

r (t) = ⟨ \sin (t^{2}), 4 t^{2} - t ⟩

.
Step 2
We find derivative of components "w.r.t" to "t"
ie

\frac{d}{d t} = \cos (t^{2}) \frac{d}{d t} (t^{2})

= \cos (t^{2}) \cdot 2 t a s k P S K ∵ \frac{d}{d t} (t^{n}) = n t^{n - 1}

\frac{d}{d t} (4 t^{2} - t) = \frac{d}{d t} (4 t^{2}) - \frac{d}{d t} (t)

= 4 \frac{d}{d t} (t^{2}) - 1

= 8 t - 1

Hence

r ‘ (t) = ⟨ 2 t \cos (t^{2}), 8 t - 1 ⟩

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