The nth derivative of 1/x is denoted by $\frac{d^n}{dx^n}(\frac{1}{x})$ and it is equal to (-1)nn!/xn+1. The nth derivative of 1/(ax+b) is denoted by $\frac{d^n}{dx^n}(\frac{1}{ax+b})$ and it is equal to (-1)nann!/(ax+b)n+1. So the n-th derivative formulas of 1/x and logx are given as follows: nth Derivative of 1/x Question: What is the nth derivative of 1/x? … Read more
The nth derivative of sinx and cosx with respect to x are equal to sin(nπ/2 +x) and cos(nπ/2 +x) respectively. In this article, let us learn how to differentiate sinx and cosx with respect to x n-times. The nth order derivative of sinx and cosx are respectively denoted by $\dfrac{d^n}{dx^n}$(sinx) and $\dfrac{d^n}{dx^n}$(cosx). So their formulas … Read more
The Leibnitz’s theorem is used to find the n-th order derivative of a product function. This theorem is a generalisation of the product rule of differentiation. In this article, we will learn about Leibnitz’s theorem and solve few problems to learn how to use this theorem. Statement of Leibnitz’s Theorem Let u and v be … Read more
The beta and gamma functions are one of the important improper integrals. There integrals converge for certain values. In this article, we will learn about beta and gamma functions with their definition of convergence, properties and some solved problems. Beta Function For integers m and n, let us consider the improper integral $\int_0^1$ xm-1 (1-x)n-1. … Read more
In this post, we will prove that the functions 1/x and sin(1/x) defined on (0, 1) are not uniformly continuous on (0, 1). 1/x is not Uniformly Continuous Question: Prove that $\dfrac{1}{x}$ is not uniformly continuous on (0, 1). Solution: If $f(x) = \dfrac{1}{x}$ is uniformly continuous on (0, 1) then for every Cauchy sequence … Read more
The function f(x) = x2 defined on (0, 1) is uniformly continuous, but it is not on (0, ∞). This will be proved in this article. x2 is Uniformly Continuous on (0, 1) Let us now prove that the function f(x) = x2 is uniformly continuous on (0, 1) using the definition of uniform continuity. … Read more
The function log x is defined for all positive values of x. Note that log(x) is uniformly continuous on [1, ∞). It is also uniformly continuous on any open interval (a, b) where both a (>0) and b are finite. log x is Uniformly Continuous Let us consider the function f(x) = log(x), x ∈ … Read more
If a function satisfies the Lipschitz condition then it is uniformly continuous. But the converse is not true. For example, f(x) = √x on [0, 1]. Definition of Lipschitz Function If a function f(x) satisfy Lipschitz condition on an interval I ⊂ ℝ then there exists a real number K > 0 such that for … Read more
A uniformly continuous function is always continuous. But the converse is not true. For example, f(x)= 1/x on (0, 2). In this post, we will provide an example which is continuous but not uniformly continuous. Before we do that let us recall their definitions. Continuous Function: A function f(x) is said to be continuous at … Read more
By the properties of real numbers, we basically mean the how various algebraic operations (eg., +, – , ×, ÷, <, > etc) work on real numbers. Some of the basic properties of real numbers are given as follows: We now provide the complete list and let us understand these properties of real numbers one … Read more