Difference between revisions of "Total derivative"
From Encyclopedia of Mathematics
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| − | The [[Derivative|derivative]] with respect to $t$ of the function $y=f(t,u_1,\ | + | The [[Derivative|derivative]] with respect to $t$ of the function $y=f(t,u_1,\dotsc,u_m)$ which depends on the variable $t$ not only directly but also via the intermediate variables $u_1=u_1(t,x_1,\dotsc,x_n),\dotsc,u_m=u_m(t,x_1,\dotsc,x_n)$. It is calculated by the formula |
| − | $$\frac{dy}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u_1}\frac{\partial u_1}{\partial t}+\ | + | $$\frac{dy}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u_1}\frac{\partial u_1}{\partial t}+\dotsb+\frac{\partial f}{\partial u_m}\frac{\partial u_m}{\partial t},$$ |
where $\partial f/\partial t$, $\partial f/\partial u_1,\dots,\partial f/\partial u_m$, $\partial u_1/\partial t,\dots,\partial u_m/\partial t$ are partial derivatives (cf. [[Partial derivative|Partial derivative]]). | where $\partial f/\partial t$, $\partial f/\partial u_1,\dots,\partial f/\partial u_m$, $\partial u_1/\partial t,\dots,\partial u_m/\partial t$ are partial derivatives (cf. [[Partial derivative|Partial derivative]]). | ||
Latest revision as of 12:41, 14 February 2020
of a composite function
The derivative with respect to $t$ of the function $y=f(t,u_1,\dotsc,u_m)$ which depends on the variable $t$ not only directly but also via the intermediate variables $u_1=u_1(t,x_1,\dotsc,x_n),\dotsc,u_m=u_m(t,x_1,\dotsc,x_n)$. It is calculated by the formula
$$\frac{dy}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u_1}\frac{\partial u_1}{\partial t}+\dotsb+\frac{\partial f}{\partial u_m}\frac{\partial u_m}{\partial t},$$
where $\partial f/\partial t$, $\partial f/\partial u_1,\dots,\partial f/\partial u_m$, $\partial u_1/\partial t,\dots,\partial u_m/\partial t$ are partial derivatives (cf. Partial derivative).
How to Cite This Entry:
Total derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_derivative&oldid=33249
Total derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_derivative&oldid=33249
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article