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Math Help - Differentiate with respect to x

  1. #1
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    Differentiate with respect to x

    Please attempt to show working where ever possible...

    See the attached file for questions....
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    Last edited by duckegg; June 15th 2009 at 02:34 PM. Reason: Spelling
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  2. #2
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    Hello, duckegg!

    Differentiate: . (a)\;\;y \;=\;\sqrt{x^3+5}

    We have: . y \;=\;\left(x^3+5\right)^{\frac{1}{2}}

    Chain Rule: . \frac{dy}{dx} \;=\;\tfrac{1}{2}(x^3+5)^{-\frac{1}{2}}\cdot 3x^2 \;=\;\frac{3x^2}{2\sqrt{x^3+5}}



    (b)\;\;y \;=\;x^3\sin2x

    Product Rule: . \frac{dy}{dx}\;=\;x^3\cdot\cos2x\cdot2 + 3x^2\cdot\sin2x \;=\;2x^3\cos2x + 3x^2\sin2x



    (c)\;\;y \:=\:\frac{e^{2x}}{x+3}

    Quotient Rule: . \frac{dy}{dx}\;=\;\frac{(x+3)\cdot e^{2x}\cdot 2 - e^{2x}\cdot 1}{(x+3)^2} \;=\;\frac{e^{2x}(2x+6-1)}{(x+3)^2} \;=\;\frac{e^{2x}(2x+5)}{(x+3)^2}

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