nth derivative of 1/x | nth derivative of 1/(ax+b)

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

nth Derivative of sinx | nth Derivative of cosx

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

Beta and Gamma Functions: Definition, Properties, Solved Problems

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

1/x and sin(1/x) are not Uniformly Continuous on (0,1)

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

Continuous but not Uniformly Continuous: An Example

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

Properties of Real Numbers

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