# Leibniz formula for derivatives

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##### 2: 1.5 Calculus of Two or More Variables
###### §1.5(i) Partial Derivatives
The function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, $\,{\partial}^{2}f/\,\partial x\,\partial y$, and $\,{\partial}^{2}f/\,\partial y\,\partial x$ are continuous. …
##### 3: 17.2 Calculus
###### §17.2(iv) Derivatives
The $q$-derivatives of $f(z)$ are defined by … When $q\to 1$ the $q$-derivatives converge to the corresponding ordinary derivatives. …
##### 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{% z(z-1)}w=0.$
31.12.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{% \gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.$
31.12.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$
31.12.4 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\gamma+z\right)z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0.$
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: 1.2 Elementary Algebra
and $f^{(k)}$ is the $k$-th derivative of $f$1.4(iii)). … where $\det(\mathbf{A})$ is defined by the Leibniz formulaFormula (1.2.77) is more generally valid for all square matrices $\mathbf{A}$, not necessarily non-defective, see Hall (2015, Thm 2.12).
##### 7: 4.24 Inverse Trigonometric Functions: Further Properties
###### §4.24(ii) Derivatives
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. …
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). …
13.3.23 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}U\left(a,b,z\right)=(-1)^{n}{\left(a% \right)_{n}}U\left(a+n,b+n,z\right),$
13.3.29 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$ $n=1,2,3,\dots$.