How do I evaluate this: $$ \frac{d}{dx}\left(\frac{dx}{dt}\right) $$ given that $x$ is function of $t$
What is derivative of a derivative of function with respect to the function?
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calculus
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1What are your thoughts, and what have you tried? Is there a specific function $x(t)$ that you're interested in? That will make this easier. – 2017-02-05
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0HINT: Apply the chain rule. – 2017-02-05
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0Do you know how what it means for a function between (possibly infinite dimensional) topological vector spaces to be differentiable? Here I would view $x\mapsto \frac{dx}{dt}$ as a map from $C^\infty(\Bbb R)\to C^\infty(\Bbb R)$, or as a map from $C^1_0(\Bbb R)\to C_0(\Bbb R)$. Is this the kind of context you are expecting? – 2017-02-05
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0Possible duplicate of [Derivative of a function with respect to another function.](http://math.stackexchange.com/questions/954073/derivative-of-a-function-with-respect-to-another-function) – 2017-02-06
1 Answers
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You will need to use the Chain Rule to express the $d/dx$ with respect to $t$.
$$ \frac{d}{dx}\left(\frac{dx}{dt}\right)=\frac{\frac{d}{dt}\left(\frac{dx}{dt}\right)}{\frac{dx}{dt}} $$