SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
Derivative
f'
represents the derivative of a function f of one argument.
Derivative[n_{1}, n_{2}, ...][f]
is the general form, representing a function obtained from f by differentiating times with respect to the first argument, times with respect to the second argument, and so on.
DetailsDetails
 f' is equivalent to Derivative[1][f].
 f'' evaluates to Derivative[2][f].
 You can think of Derivative as a functional operator which acts on functions to give derivative functions.
 Derivative is generated when you apply D to functions whose derivatives Mathematica does not know.
 Mathematica attempts to convert Derivative[n][f] and so on to pure functions. Whenever Derivative[n][f] is generated, Mathematica rewrites it as D[f[#], {#, n}]&. If Mathematica finds an explicit value for this derivative, it returns this value. Otherwise, it returns the original Derivative form.
 Derivative[n][f] represents the indefinite integral of f.
 Derivative[{n_{1}, n_{2}, ...}][f] represents the derivative of taken times with respect to . In general, arguments given in lists in f can be handled by using a corresponding list structure in Derivative.
 N[f'[x]] will give a numerical approximation to a derivative.
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