Use the chain rule to find the derivative of each of the following : a. [tex]\frac{2}{(3x^2-x+7)^3}[/tex] b. [tex](4x-10)^

Use the chain rule to find the derivative of each of the following :
a. \frac{2}{(3x^2-x+7)^3}

b. (4x-10)^1^0

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  1. Answer:

    see explanation

    Step-by-step explanation:

    Using the chain rule

    Given

    y = f(g(x)), then

    [tex]\frac{dy}{dx}[/tex] = f'(g(x)) × g'(x)

    (a)

    y = [tex]\frac{2}{(3x^2-x+7)^3}[/tex] = 2 ([tex]3x^2-x+7)^{-3}[/tex]

    [tex]\frac{dy}{dx}[/tex] = 2. – 3 [tex](3x^2-x+7)^{-4}[/tex] × [tex]\frac{d}{dx}[/tex] (3x² – x + 7)

       = – 6 [tex](3x^2-x+7)^{-4}[/tex] × (6x – 1)

       = [tex]\frac{-6(6x-1)}{(3x^2-x+7)^4}[/tex]

    —————————————————

       (b)

    y = [tex](4x-10)^{10}[/tex]

    [tex]\frac{dy}{dx}[/tex] = 10[tex](4x-10)^{9}[/tex] × [tex]\frac{d}{dx}[/tex](4x – 10)

        = 10[tex](4x-10)^{9}[/tex] × 4

         = 40[tex](4x-10)^{9}[/tex]

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