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[tex]f(x(t),y(t))=2(x(t))^2+x(t)y(t)+y(t)[/tex]

One way to calculate df/dt is directly using the chain rule:

[tex]\frac{df}{dt}=4x(t)\frac{dx}{dt}+\frac{dx}{dt}y(t)+\frac{dy}{dt}x(t)+\frac{dy}{dt}[/tex]

[tex]\frac{df}{dt}=(4x(t)+y(t))\frac{dx}{dt}+(x(t)+1)\frac{dy}{dt}[/tex]

Another way is by using the formula for the total derivative:

[tex]\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}[/tex]

[tex]\frac{\partial f}{\partial x} = 4x+y[/tex]

[tex]\frac{\partial f}{\partial y} = x+1[/tex]

[tex]\frac{df}{dt} = (4x+y)\frac{dx}{dt}+(x+1)\frac{dy}{dt}[/tex]

I see how the formula for total derivatives should work since it is the multivariable analog of the derivative, but is there any way to logically derive formula for the total derivative from single variable calculus (just using chain rule, product rule, etc.), without having to visualize things in "3D"? It seems like you should be able to since f(x(t),y(t))=f(t) is really just a single variable function.

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# I How to logically derive the total derivative formula?

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