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Given is the function

f = f(a,b,t), where a=a(b) and b = b(t). Need to express first and second order derivatives.

[itex]\frac{\partial f}{\partial a}[/itex] and [itex]\frac{\partial f}{\partial b}[/itex] should be just that, nothing more to it here, correct?

But

[itex]\frac{df}{dt}[/itex] = [itex]\frac{\partial f}{\partial a}[/itex] [itex]\frac{da}{db}[/itex] [itex]\frac{db}{dt}[/itex] + [itex]\frac{\partial f}{\partial b}[/itex] [itex]\frac{db}{dt}[/itex] + [itex]\frac{\partial f}{\partial t}[/itex], by the chain rule, correct?

I need to express [itex]\frac{\partial f}{\partial t}[/itex], but the above chain rule puts the total derivative [itex]\frac{df}{dt}[/itex] in the expression and it gets messy. I mean, how do I express

[itex]\frac{\partial f}{\partial t}[/itex]?

Then I need also [itex]\frac{\partial^2 f}{\partial t^2}[/itex], [itex]\frac{\partial^2 f}{\partial a^2}[/itex] and [itex]\frac{\partial^2 f}{\partial b^2}[/itex].

Anyone well versed in partial derivatives?

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# Multivariable Chain rule for higher order derivatives

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