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Math Help - [SOLVED] Substitution help...

  1. #1
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    [SOLVED] Substitution help...

    Hi, I just need help with taking the derivative of the substition factor. From there I can (hopefully) get the rest.

    DE:
    x^3 \dfrac{dy}{dx} + x^2y+x=3e^{xy}

    Change of variable given in problem:
    z=e^{xy}

    Is this part right, usually the right side of this only has a y, so im not quite sure what to do. Thanks.
    \dfrac{d}{dx}[z=e^{xy}]

    \dfrac{dz}{dx}=ye^{xy}\dfrac{dy}{dx}
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  2. #2
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    Quote Originally Posted by snaes View Post
    Hi, I just need help with taking the derivative of the substition factor. From there I can (hopefully) get the rest.

    DE:
    x^3 \dfrac{dy}{dx} + x^2y+x=3e^{xy}

    Change of variable given in problem:
    z=e^{xy}

    Is this part right, usually the right side of this only has a y, so im not quite sure what to do. Thanks.
    \dfrac{d}{dx}[z=e^{xy}]

    \dfrac{dz}{dx}=ye^{xy}\dfrac{dy}{dx}
    if z = e^{xy}, then \frac{dz}{dx} = e^{xy}\left(y + x\frac{dy}{dx}\right)
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  3. #3
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    ahh

    ahh with respect to both x and y.
    THANKS!
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  4. #4
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    Quote Originally Posted by snaes View Post
    ahh with respect to both x and y.
    THANKS!
    No, not with respect to both x and y. Only with respect to x. But since y is a function of x, you have to use the chain rule.


    I.e. z = e^{xy}

    Let u = xy so that z = e^u.


    \frac{du}{dx} = x\frac{d}{dx}(y) + y\frac{d}{dx}(x)

     = x\frac{dy}{dx} + y.


    \frac{dz}{du} = e^u = e^{xy}.


    So \frac{dz}{dx} = \left(x\frac{dy}{dx} + y\right)e^{xy}.
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