"Find the derivative of the vector function. r(t)=et2i−j+ln⁡(1+3t)k" was answered by our community

gabioskay7

Answered question

2021-12-04

Find the derivative of the vector function.

r (t) = e^{t^{2}} i - j + \ln (1 + 3 t) k

Answer & Explanation

George Burge

Beginner2021-12-05Added 16 answers

To find the derivative of

r (t) = e^{t^{2}} i - j + \ln (1 + 3 t) k

we will use Theorem 2 and Theorem 3 (differentiation rules). We have:

\frac{d r (t)}{d t} = {(e^{t^{2}})}^{'} i - 1^{'} j + {[\ln (1 + 3 t)]}^{'} k

= e^{t^{2}} \cdot {(t^{2})}^{'} i - 0 j + \frac{1}{1 + 3 t} \cdot {(1 + 3 t)}^{'} k

= 2 t e^{t^{2}} i + \frac{3}{1 + 3 t} k

where we used the chain rule for the first and third component. We conclude that

\frac{d r (t)}{d t} = 2 t e^{t^{2}} i + \frac{3}{1 + 3 t} k

Result:

\frac{d r (t)}{d t} = 2 t e^{t^{2}} i + \frac{3}{1 + 3 t} k

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