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Math Help - Chain rule - 2 variables

  1. #1
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    Chain rule - 2 variables

    Hello MHF,
    Would appreciate some help with the following question,
    Find \frac{dz}{dt} if z=txy^2 ,\   x=t+\ln(y+t^2)\ \mbox{and}\ y=e^2. You may express your answer as a function of t,x,y.
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  2. #2
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    Quote Originally Posted by Robb View Post
    Hello MHF,
    Would appreciate some help with the following question,
    Find \frac{dz}{dt} if z=txy^2 ,\ x=t+\ln(y+t^2)\ \mbox{and}\ y=e^2. You may express your answer as a function of t,x,y.
    Using some substitutions

    z=txy^2 becomes z=t(t+\ln(e^2+t^2))(e^2)^2

    Now you only have to find \frac{dz}{dt} with respect to 1 variable. I would expand from here and differentiate each term separately.
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  3. #3
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    In general: \frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt} if z(x,y)=z(x(t),y(t))
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  4. #4
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    Actually, here, your original function z is a function of three variables, x, y, and t.

    The chain rule for this would be \frac{dz}{dt}= \frac{\partial z}{\partial t}+ \frac{\partial z}{\partial x}\frac{dx}{dt}+ \frac{\partial z}{\partial y}\frac{dy}{dt}

    The \frac{\partial z}{\partial t}, of course, would just be xy^2.
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  5. #5
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    Good catch, I didn't notice that the function was made of three variables.
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