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I'm struggling to find the derivative of a function with multiple inputs, but some inputs are also inputs to other variables. In this case, both t and x(t) are passed in to f.

Given: $f(x, \dot x, t) = 10x\dot x$ where $\dot x=\frac{dx}{dt}$, find $\frac{\partial f}{\partial t}$

Steps so far: $$\frac{\partial}{\partial t}(f) = \frac{\partial}{\partial t}(10x\dot x)$$ $$\frac{\partial f}{\partial t} = 10\frac{\partial}{\partial t}(x\dot x)$$

Product rule between $x$ and $\dot x$ would be the next step. The part that I'm struggling with is taking $\frac{\partial}{\partial t}(x)$ and $\frac{\partial}{\partial t}(\dot x)$. Intuitively (to me), $\frac{\partial}{\partial t}(x) = \dot x$, but $\frac{d}{dt}(x) = \dot x$ as well. Am I getting confused with notation, or is there a part of partial derivatives that I don't understand? Thanks.

Elias Benevedes
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  • Since $x$ only depends on $t$, $\frac \partial {\partial t} (x)$ and $\frac d {dt} (x)$ are equivalent. In other words, the $\frac d {dt}$ operator is the special case where the operator $\frac \partial {\partial t}$ is operating on only depends on $t$. – player3236 Sep 05 '20 at 19:15

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$$\frac{\partial}{\partial t}f(x, \dot x, t) = \frac{d}{d t}(10x\dot x)= 10\left(\left(\frac{dx}{dt}\right)^2+ x\frac{d^2x}{dt^2}\right)$$

zkutch
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