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Math Help - product rule

  1. #1
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    Exclamation product rule

    Dervative of (3x+2)^4 (5x+2)^-5
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Nimmy View Post
    Dervative of (3x+2)^4 (5x+2)^-5
    Use the product rule. Call f(x) = (3x+2)^4 and g(x) = (5x+2)^{-5}

    Then f'(x) = 4(3x+2)^3 \cdot 3 = 12(3x+2)^3.

    And g'(x) = -5(5x+2)^{-6} \cdot 5 = -25(5x+2)^{-6}.

    Thus the derivative of (3x+2)^4 (5x+2)^{-5} is f'(x)g(x) + f(x)g'(x)

    = 12(3x+2)^3 \cdot (5x+2)^{-5} + (3x+2)^4 \cdot -25(5x+2)^{-6}

    = 12(3x+2)^3(5x+2)^{-5} - 25(3x+2)^4(5x+2)^{-6}

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by topsquark View Post
    Use the product rule. Call f(x) = (3x+2)^4 and g(x) = (5x+2)^{-5}

    Then f'(x) = 4(3x+2)^3 \cdot 3 = 12(3x+2)^3.

    And g'(x) = -5(5x+2)^{-6} \cdot 5 = -25(5x+2)^{-6}.

    Thus the derivative of (3x+2)^4 (5x+2)^{-5} is f'(x)g(x) + f(x)g'(x)

    = 12(3x+2)^3 \cdot (5x+2)^{-5} + (3x+2)^4 \cdot -25(5x+2)^{-6}

    = 12(3x+2)^3(5x+2)^{-5} - 25(3x+2)^4(5x+2)^{-6}

    -Dan
    If you need that in fraction form:

    (Continuing)

    = \frac{12(3x+2)^3}{(5x+2)^5} - \frac{25(3x+2)^4}{(5x+2)^6}

    = \frac{12(3x+2)^3}{(5x+2)^5} \cdot \frac{5x+2}{5x+2} - \frac{25(3x+2)^4}{(5x+2)^6}

    = \frac{12(3x+2)^3(5x+2) - 25(3x+2)^4}{(5x+2)^6}

    = \frac{(3x+2)^3(12(5x+2) - 25(3x+2))}{(5x+2)^6}

    = \frac{(3x+2)^3(60x + 24 - 75x - 50)}{(5x+2)^6}

    = \frac{(3x+2)^3(-15x - 26)}{(5x+2)^6}

    = -\frac{(3x+2)^3(15x + 26)}{(5x+2)^6}

    -Dan
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Nimmy View Post
    Dervative of (3x+2)^4 (5x+2)^-5
    Or you can use the quotient rule. Let f(x) = (3x+2)^4 and g(x) = (5x+2)^5.

    Then f'(x) = 4(3x+2)^3 \cdot 3 = 12(3x+2)^3.

    And g'(x) = 5(5x+2)^4 \cdot 5 = 25(5x+2)^4.

    Then the derivative of \frac{(3x+2)^4}{(5x+2)^5} is \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

    = \frac{12(3x+2)^3 \cdot (5x+2)^5 - (3x+2)^4 \cdot 25(5x+2)^4}{ \left ( (5x+2)^5 \right ) ^2}

    = \frac{(5x+2)^4(12(3x+2)^3 \cdot (5x+2) - 25(3x+2)^4)}{(5x+2)^10}

    = \frac{12(3x+2)^3(5x+2) - 25(3x+2)^4}{(5x+2)^6}

    which is the same as the third line in my second post.

    -Dan
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