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Math Help - Partial Derivative

  1. #1
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    Partial Derivative

    How would I do this partial derivative?

     \frac {d^2z}{dydx} ...........z= \frac{(x+y)^{y+1}}{y+1} +sinh^3(y)+2


    Thanks.
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  2. #2
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    Hello,

    Firstly, derivate in function of x.

    The derivative of \frac{(x+y)^{y+1}}{y+1} is (x+y)^y (do you know why ?)

    The derivative of sinh^3(y)+2 is 0 because there is no x in it, so it's like a constant if the variable is x.


    Now remains (x+y)^y to derivate in function of y.

    (x+y)^y=e^{y \ln(x+y)}

    The derivative for e^{f(x)} is f'(x)e^{f(x)}

    Here, f(y)=y \ln(x+y). So, by using the product rule, we have : f'(y)=\frac{y}{x+y}+\ln(x+y)


    Hence \frac{d^2z}{dydx}=e^{y \ln(x+y)}(\frac{y}{x+y}+\ln(x+y)) =(x+y)^y \frac{y}{x+y}+(x+y)^y \ln(x+y)=y(x+y)^{y-1}+(x+y)^y \ln(x+y)
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