Several Differentiation Formulas of Special Functions. Part IV
نویسندگان
چکیده
The notation and terminology used here are introduced in the following papers: [13], [15], [1], [16], [2], [4], [10], [11], [17], [5], [14], [12], [3], [7], [6], [9], and [8]. For simplicity, we adopt the following convention: x, a, b, c denote real numbers, n denotes a natural number, Z denotes an open subset of R, and f , f1, f2 denote partial functions from R to R. Next we state a number of propositions: (1) If x ∈ dom (the function tan), then (the function cos)(x) 6= 0. (2) If x ∈ dom (the function cot), then (the function sin)(x) 6= 0. (3) If Z ⊆ dom( f1 f2 ), then for every x such that x ∈ Z holds ( f1 f2 )(x) Z = f1(x)nZ f2(x)nZ . (4) Suppose Z ⊆ dom( f1 f2 ) and for every x such that x ∈ Z holds f1(x) = x + a and f2(x) = x − b. Then f1 f2 is differentiable on Z and for every x such that x ∈ Z holds ( f1 f2 )′↾Z(x) = −a−b (x−b) . (5) Suppose Z ⊆ dom((the function ln) · 1 f ) and for every x such that x ∈ Z holds f(x) = x. Then (the function ln) · 1 f is differentiable on Z and for every x such that x ∈ Z holds ((the function ln) · 1 f )′↾Z(x) = − 1 x .
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