- #1
Sunfire
- 221
- 4
Hello,
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?
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?