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is there a difference between $\frac{\partial^2 }{\partial x^2}$ and $(\frac{\partial }{\partial x})^{2}$? I have to tell if a differential equation is linear, and $(\frac{\partial }{\partial x})^{2}$ concerns me. If it's the same thing as second derivative, the equation would be linear.

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Same thing; it means the operator $\partial / \partial x$ applied twice. –  anon Apr 1 '12 at 20:27
    
Is the notation $\frac{\partial^2}{\partial x^2}$ also used in the same text? If so, that would seem to indicate that the author intends for $(\frac{\partial}{\partial x})^2$ to mean something different, in which case Tom Au's answer below seems to be correct. –  Santiago Canez Jan 30 at 23:00
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2 Answers

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the notation $\frac{\partial^2 }{\partial x^2}f~~$ means $\frac{\partial }{\partial x}(\frac{\partial }{\partial x}f)$, so $\frac{\partial^2 }{\partial x^2}f~~$ and $(\frac{\partial }{\partial x})^{2}f~~$ is the same both you apply $\frac{\partial }{\partial x}$ twice.

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The first expression, $\frac{\partial^2 }{\partial x^2}$ is the second derivative, while the second expression, $(\frac{\partial }{\partial x})^{2}$, is the first derivative squared.

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