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In the p-adic setting, the usual definition of the derivative fails to have certain desirable properties. For instance, it is possible for a function that is not locally constant to have zero derivative everywhere. An example of this is furnished by the function F: Z p → Z p, where Z p is the ring of p-adic integers, defined by
For any functions and and any real numbers and , the derivative of the function () = + with respect to is ′ = ′ + ′ (). In Leibniz's notation , this formula is written as: d ( a f + b g ) d x = a d f d x + b d g d x . {\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.
For all other values of , the expression is not well-defined for <, as was covered above, or is not a real number, so the limit does not exist as a real-valued derivative. For the two cases that do exist, the values agree with the value of the existing power rule at 0, so no exception need be made.
The method returns a pair of the evaluated function and its derivative. The method traverses the expression tree recursively until a variable is reached. If the derivative with respect to this variable is requested, its derivative is 1, 0 otherwise. Then the partial function as well as the partial derivative are evaluated. [16]
But the zero function is not a weak derivative of c, as can be seen by comparing against an appropriate test function . More theoretically, c does not have a weak derivative because its distributional derivative , namely the Cantor distribution , is a singular measure and therefore cannot be represented by a function.
for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
To obtain more general derivative approximation formulas for some function (), let > be a positive number close to zero. The Taylor expansion of f ( x ) {\displaystyle f(x)} about the base point x {\displaystyle x} is