Find the derivative r ‘(t) of the vector function r(t). t sin 6t , t2, t cos 7t Part 1 of 4 The derivative of a vector function is obtained

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Find the derivative r ‘(t) of the vector function r(t). t sin 6t , t2, t cos 7t Part 1 of 4 The derivative of a vector function is obtained by differentiating each of its components. Thus, if r(t) = f(t), g(t), h(t) , where f, g, and h are differentiable functions, then r'(t) = f ‘(t), g'(t), h'(t) . For r(t) = t sin 6t, t2, t cos 7t , we have f(t) = t sin 6t, which is a product. Using the Product Rule and the Chain Rule, we have

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RI SƠ 3 months 2021-08-02T18:33:10+00:00 1 Answers 4 views 0

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    2021-08-02T18:34:34+00:00

    Answer:

    r'(t)=(sin(6t)+6tcos(6t),2t,cos(7t)-7tsin(7t))

    Step-by-step explanation:

    We need to find the derivative r'(t) of the vector function :

    r(t)=(tsin(6t),t^{2},tcos(7t))

    In order to find r'(t), we are going to differentiate each of its components ⇒

    We can write the following ⇒

    r(t)=(f(t),g(t),h(t))=(tsin(6t),t^{2},tcos(7t))

    f(t)=tsin(6t)\\g(t)=t^{2}\\h(t)=tcos(7t)

    Let’s differentiate each function to obtain r'(t) :

    f(t)=tsin(6t)f'(t)=1.sin(6t)+t.cos(6t).6=sin(6t)+6tcos(6t)

    f'(t)=sin(6t)+6tcos(6t)

    Now with g(t) :

    g(t)=t^{2}

    g'(t)=2t

    With h(t) :

    h(t)=tcos(7t)h'(t)=1.cos(7t)+t[-sin(7t)].7

    h'(t)=cos(7t)-7tsin(7t)

    Finally we need to complete r'(t)=(f'(t),g'(t),h'(t)) with its components :

    r'(t)=(sin(6t)+6tcos(6t),2t,cos(7t)-7tsin(7t))

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