"Derivatives of vector-valued functions Differentiate the following function...." was answered by our community

Halkadvalseln

Answered question

2021-11-26

Derivatives of vector-valued functions
Differentiate the following function.

r (t) = ⟨ 4, 3 \cos 2 t, 2 \sin 3 t ⟩

Mary Ramirez

Beginner2021-11-27Added 19 answers

Step 1
The given function is:

r (t) = ⟨ 4, 3 \cos (2 t), w \sin (3 t) ⟩

We have to differentiate the given function.
The given function can be written as:

r (t) = 4 i + 3 \cos (2 t) j + 2 \sin (3 t) k

Step 2
Therefore we have

r (t) = 4 i + 3 \cos (2 t) j + 2 \sin (3 t) k

Now differentiate both the sides with respect to t.
Therefore,

\frac{d (r (t))}{d t} = \frac{d (4 i + 3 \cos (2 t) j + 2 \sin (3 t) k)}{d t}

= \frac{d 4}{d t} i + \frac{d 3 \cos (2 t)}{d t} j + \frac{d 2 \sin (3 t)}{d t} k

= 0 i + 3 \frac{d \cos (2 t)}{d t} j + 2 \frac{d \sin (3 t)}{d t} k

= 3 (- \sin (2 t)) \frac{d 2 t}{d t} j + 2 (\cos (3 t)) \frac{d 3 t}{d t} k

= - 3 \sin (2 t) \times 2 j + 2 \cos (3 t) \times 3 k

= - 6 \sin (2 t) j + 6 \cos (3 t) k

Step 3
Therefore,

\frac{d (r (t))}{d t} = - 6 \sin (2 t) j + 6 \cos (3 t) k

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