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Math Help - differentiation of exponential function

  1. #1
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    differentiation of exponential function

    Hello, I am stuck on differentiating this:

    6 * e^(x^(1/2))

    Can someone please show me the steps. I understand chain and product rules will be implemented but I cannot seem to get it working.

    Thanks kindly for any help.
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  2. #2
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    Re: differentiation of exponential function

    Quote Originally Posted by fran1942 View Post
    Hello, I am stuck on differentiating this:

    6 * e^(x^(1/2))

    Can someone please show me the steps. I understand chain and product rules will be implemented but I cannot seem to get it working.

    Thanks kindly for any help.
    \displaystyle \begin{align*} y = 6e^{x^{\frac{1}{2}}} \end{align*}

    Let \displaystyle \begin{align*} u = x^{\frac{1}{2}} \end{align*} so that \displaystyle \begin{align*} y = 6e^u \end{align*} , then


    \displaystyle \begin{align*} \frac{du}{dx} = \frac{1}{2}x^{-\frac{1}{2}} \end{align*}


    \displaystyle \begin{align*} \frac{dy}{du} &= 6e^u \\ &= 6e^{x^{\frac{1}{2}}} \end{align*}

    So

    \displaystyle \begin{align*} \frac{dy}{dx} &= \frac{du}{dx} \cdot \frac{dy}{du} \\ &= \frac{1}{2}x^{-\frac{1}{2}} \cdot 6e^{x^{\frac{1}{2}}} \\ &= 3x^{-\frac{1}{2}}e^{x^{\frac{1}{2}}} \end{align*}
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  3. #3
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    Re: differentiation of exponential function

    Quote Originally Posted by fran1942 View Post
    Hello, I am stuck on differentiating this:

    6 * e^(x^(1/2))

    Can someone please show me the steps. I understand chain and product rules will be implemented but I cannot seem to get it working.

    Thanks kindly for any help.
    y' = {\left( 6 \right)^'} \cdot {{\rm{e}}^{\sqrt x }} + 6 \cdot {\left( {{{\rm{e}}^{\sqrt x }}} \right)^'} = 6 \cdot {\left( {{{\rm{e}}^{\sqrt x }}} \right)^'} = 6 \cdot {{\rm{e}}^{\sqrt x }} \cdot {\left( {\sqrt x } \right)^'} = \frac{3}{{\sqrt x }} \cdot {{\rm{e}}^{\sqrt x }}
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