If $z=z(x,y)$, and $x=x(t),y=y(t)$, write down the expression for $\frac{dz}{dx}$. I am confused by the difference between partial and regular derivatives when they are used together.
Partial derivatives turning into regular derivative
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$\begingroup$
derivatives
partial-derivative
1 Answers
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Observe that $$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$ $$\implies \frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dt}\cdot \frac{dt}{dx}$$ $$\implies \frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\cdot \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
Tell me if you don't get the first line.
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0Yes, I am new to this and I don't understand the first line. – 2017-02-21
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0Read the part on *Total Differential* from this link ... http://www.solitaryroad.com/c353.html – 2017-02-21
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0Awesome, thank you very much! – 2017-02-21
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0You're welcome. – 2017-02-21
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0I have just noticed that I was actually asked for $\frac{dz}{dt}$. Should I just divide the total differential by $dt$? – 2017-02-21
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0Yes, you need that only. – 2017-02-21