1.5.2 Differentiation| Mathematica knows the derivatives of all the standard mathematical functions. | |
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D[ ArcTan[x], x ]
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| This differentiates three times with respect to x. | |
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The function D[x^n, x] really gives a partial derivative, in which n is assumed not to depend on x. Mathematica has another function, called Dt, which finds total derivatives, in which all variables are assumed to be related. In mathematical notation, D[f, x] is like  , while Dt[f, x] is like  . You can think of Dt as standing for "derivative total". Dt gives a total derivative, which assumes that n can depend on x. Dt[n, x] stands for  . | |
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| D[f, x] | partial derivative  | D[f,  ,  , ... ] | multiple derivative  | | D[f, {x, n}] | repeated derivative  | | Dt[f] | total differential  | | Dt[f, x] | total derivative  |
Some differentiation functions. As well as treating variables like  symbolically, you can also treat functions in Mathematica symbolically. Thus, for example, you can find formulas for derivatives of f[x], without specifying any explicit form for the function f. | Mathematica does not know how to differentiate f, so it gives you back a symbolic result in terms of f'. | |
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| Mathematica uses the chain rule to simplify derivatives. | |
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D[ 2 x f[x^2], x ]
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with respect to 
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