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Leibniz formula for derivatives

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1: 1.4 Calculus of One Variable
§1.4(iii) Derivatives
Higher Derivatives
Chain Rule
Leibniz’s Formula
Faà Di Bruno’s Formula
2: 1.5 Calculus of Two or More Variables
§1.5(i) Partial Derivatives
The function f ( x , y ) is continuously differentiable if f , f / x , and f / y are continuous, and twice-continuously differentiable if also 2 f / x 2 , 2 f / y 2 , 2 f / x y , and 2 f / y x are continuous. …
Chain Rule
§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
§1.5(vi) Jacobians and Change of Variables
3: 17.2 Calculus
§17.2(iv) Derivatives
The q -derivatives of f ( z ) are defined by … When q 1 the q -derivatives converge to the corresponding ordinary derivatives. …
Leibniz Rule
4: 36.4 Bifurcation Sets
§36.4(i) Formulas
K = 2 , cusp bifurcation set: … K = 3 , swallowtail bifurcation set: … Elliptic umbilic bifurcation set (codimension three): for fixed z , the section of the bifurcation set is a three-cusped astroid … Hyperbolic umbilic bifurcation set (codimension three): …
5: 31.12 Confluent Forms of Heun’s Equation
31.12.1 d 2 w d z 2 + ( γ z + δ z - 1 + ϵ ) d w d z + α z - q z ( z - 1 ) w = 0 .
31.12.2 d 2 w d z 2 + ( δ z 2 + γ z + 1 ) d w d z + α z - q z 2 w = 0 .
31.12.3 d 2 w d z 2 - ( γ z + δ + z ) d w d z + α z - q z w = 0 .
31.12.4 d 2 w d z 2 + ( γ + z ) z d w d z + ( α z - q ) w = 0 .
For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000). …
6: 4.24 Inverse Trigonometric Functions: Further Properties
§4.24(ii) Derivatives
4.24.7 d d z arcsin z = ( 1 - z 2 ) - 1 / 2 ,
4.24.8 d d z arccos z = - ( 1 - z 2 ) - 1 / 2 ,
4.24.9 d d z arctan z = 1 1 + z 2 .
§4.24(iii) Addition Formulas
7: 3.4 Differentiation
Two-Point Formula
Three-Point Formula
For corresponding formulas for second, third, and fourth derivatives, with t = 0 , see Collatz (1960, Table III, pp. 538–539). For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). … Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. …
8: 31.18 Methods of Computation
Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …
9: 14.7 Integer Degree and Order
§14.7(ii) Rodrigues-Type Formulas
14.7.10 P n m ( x ) = ( - 1 ) m + n ( 1 - x 2 ) m / 2 2 n n ! d m + n d x m + n ( 1 - x 2 ) n .
14.7.13 P n ( x ) = 1 2 n n ! d n d x n ( x 2 - 1 ) n ,
14.7.14 P n m ( x ) = ( x 2 - 1 ) m / 2 2 n n ! d m + n d x m + n ( x 2 - 1 ) n ,
§14.7(iii) Reflection Formulas
10: 13.3 Recurrence Relations and Derivatives
§13.3 Recurrence Relations and Derivatives
§13.3(ii) Differentiation Formulas
13.3.22 d d z U ( a , b , z ) = - a U ( a + 1 , b + 1 , z ) ,
13.3.23 d n d z n U ( a , b , z ) = ( - 1 ) n ( a ) n U ( a + n , b + n , z ) ,
13.3.29 ( z d d z z ) n = z n d n d z n z n , n = 1 , 2 , 3 , .