导函数

The formulas calculating higher derivatives of parametric functions and composite functions are given by the chain rule and the product formula for derivatives.

用函数的链式法则和乘积公式给出了参数函数和复合函数的高阶导数的计算公式。

In presented method, the correlated variables are transformed into uncorrelated variables equivalently, and the performance functions in the correlated variables space are transformed into those in the equivalent uncorrelated variables space, in which the reliability sensitivity of the system failure probability with respect to the distribution parameters of the uncorrelated variables can be solved by the moment method for the uncorrelated variables. Finally the equivalent relationship between the correlated variables and the uncorrelated variables, the chain rule for derivative are employed to obtain the reliability sensitivity of the system failure probability with respect to the means of the correlated variables.

在所提的方法中，首先将相关变量空间等效转化为不相关变量空间，并将系统多个失效模式的功能函数转化为不相关变量空间的功能函数，进而利用矩方法得到多模式系统失效概率对不相关变量空间基本分布参数的灵敏度，最后利用相关变量空间与不相关变量空间的等价关系及复合函数求导法则，求得多模式系统失效概率对相关变量空间中变量均值的灵敏度。

Then in combination with its concavo-convex function,a simple and direct proof can also be found by using theorem deduction.

通过给出关于凹凸函数的一个性质定理及其推论，对一些特定类型的三角不等式通过构造辅助函数，求出函数的二阶导数；再结合其凹凸性利用定理的推论给予简捷的证明。

Firstly, the approach formulates a cost functional to turn the inverse problem into a constrained minimization problem according to least squares criterion, then the resulting constrained minimization problem is transformed into an unconstrained minimization problem by using a penalty function technique, and then the closed Fr chet derivatives of the Lagrange function with respect to the properties are derived based on the calculus of variations, finally, one can solve the resulting problem by using any gradient-based algorithm and the finite-difference time-domain method.

该方法首先以最小二乘准则构造目标函数，将逆问题表示为约束最小化问题；接着应用罚函数法转化为无约束最小化问题；然后基于变分计算导出闭式的拉格朗日函数关于特征参数的Fr chet导数；最后借助梯度算法和时域有限差分法迭代反演德拜模型参数。

Continuity on first derivative of wave function is discussed by means of several instances of finite and infinite transition s of potential function and hereby continuity conditions is pinpointed.

结合势函数有限跃变和无限跃变的几种实例，讨论波函数一阶导数在势函数跃变处的连续性问题，从而进一步明确连续性条件。

The paper studied how to structure a kind of counterexample function in the analysis and widespread application emphatically, further pointed out the misunderstandings easily appeared in the concept and theorem of continuity, derivable, integrable, and other analytic properties of the function and gave the corresponding counter-example finally.

研究了一类函数在分析学相关反例构造中的应用，进一步指出有关函数的连续、可导、可积等分析性质的概念与定理中容易出现的误区，并给出相应的反例，希望能引起高校师生对该类函数的重视，进一步提高教学效果。

This text will talk about some new methods about the limit of disjunction function in the boundary and the operation of lead number and definite integral.

分段函数是函数问题中难点，本文就分段函数在分界点的极限，导数、定积分的运算问题探讨一些新方法。

Theorem 1 Suppose the function is differentiable at,then its partial derivative at exist, and the total differential of

定理1如果函数在点可微分，则该函数在点的偏导数必定存在，且函数在点的全微分为。

Although the merit function introduced here is the sum of squares of overdetermined equations, by using the special structure of it, we successfully form the Newton equation by avoiding the product of Jacobi matrices and using only the Jacobi matrix of the function defining NCP.

对非线性互补问题，利用所构造价值函数具有的特殊非线性最小二乘结构，成功实现了具有一般意义的构造Newton方程时仅利用价值函数的一阶导数且无须计算相应残量函数Jacobi矩阵乘积的Newton法。

Robust design objective is always structural response′s first derivative, constraint functions are nonlinear functi...

鲁棒设计目标函数为结构响应的一阶导数，约束函数也是非线性隐函数，因此是多变量的非线性优化问题。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。