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

High School
Calculus and Analysis
Calculus 2
Parametric equations

balff1t

Answered question

2021-11-29

Derivatives of vector-valued functions
Differentiate the following function.

r (t) = ⟨ {(t + 1)}^{- 1}, \tan^{- 1} t, \ln (t + 1) ⟩

Answer & Explanation

Vaing1990

Beginner2021-11-30Added 16 answers

Step 1
Differentiating with respect to t, we get
$r^{'} (t) = ⟨ \frac{d [{(t + 1)}^{- 1}]}{d x}, \frac{d (\tan^{- 1} t)}{d t}, \frac{d [\ln (t + 1)]}{d t} ⟩$
$= ⟨ - 1 \times {(t + 1)}^{- 1 - 1}, \frac{1}{1 + t^{2}}, \frac{1}{t + 1} \times \frac{d (t + 1)}{d t} ⟩$
$[∵ \frac{d (x^{n})}{d x} = n x^{n - 1}, \frac{d (\tan^{- 1} x)}{d x} = \frac{1}{1 + x^{2}}, \frac{d (\ln x)}{d x} = \frac{1}{x}]$
$= ⟨ - {(t + 1)}^{- 2}, \frac{1}{1 + t^{2}}, \frac{1}{t + 1} \times (1 + 0) ⟩$
$= ⟨ - {(t + 1)}^{- 2}, \frac{1}{1 + t^{2}}, \frac{1}{t + 1} ⟩$

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