The derivative of x is equal to 1. The function f(x)=x denotes the identity function on the set of real numbers. In this post, we will find the derivative of x by different methods; for example, by power rule, by the limit definition of derivatives. Derivative of x by Power Rule We will use the … Read more
The derivative of x2 (x square) by first principle is 2x. The function x2 denotes the square of x. In this post, we will find the derivative of x square. The derivative of x square is denoted by d/dx (x2), and its formula is given below: $\dfrac{d}{dx}$ (x2) = 2x. Derivative of x2 by First … Read more
The derivative of xe-x is equal to (1-x)e-x. In this post, we will find the derivative of xe^-x by the product rule and the first principle of derivatives. Derivative of xe-x by First Principle The derivative of $f(x)=xe^{-x}$by the first principle is given by $(xe^{-x})’$ = limh→0 $\dfrac{(x+h)e^{-(x+h)}-xe^{-x}}{h}$ = limh→0 $\dfrac{xe^{-(x+h)}+he^{-(x+h)}-xe^{-x}}{h}$ = limh→0 $\big(\dfrac{xe^{-(x+h)}-xe^{-x}}{h}$ $+\dfrac{he^{-(x+h)}}{h} … Read more
The derivative of 1 by x square is equal to -2/x3. In this blog post, we will learn how to find the differentiation of 1/x2. What is the Derivative of 1/x2? Let us now find the derivative of 1/x^2 by the power rule of derivatives. The power rule says that $\dfrac{d}{dx}(x^n)=nx^{n-1}$ Step 1: First,we write … Read more
The derivative of 1 by 1+x^2 is equal to -2x/(1+x^2)^2. In this post, we find the derivative of 1/(1+x^2) using the chain rule and substitution method. Derivative of 1/(1+x^2) Let us find the derivative of 1/(1+x^2) by the substitution method and the product rule of derivatives. Suppose that $u=\dfrac{1}{1+x^2}$ …(I) We will need to find … Read more
The function e to the power x2 is written as $e^{x^2}$ and its derivative is $2xe^{x^2}$. In this post, we will find the derivative of e to the power x square by the first principle and chain rule of derivatives. Recall the first principle of derivatives: The derivative of a function f(x) by first principle … Read more
The derivative of 1 by root x is -1/(2x root(x)). In this post, we will find the derivative of 1/root(x) by the first principle of derivatives. We will also find the integration of 1 by root x. Derivative of 1 by root x from First Principle The first principle of derivatives says that the derivative … Read more
The function cos square x is the product of two cosine functions. The derivative of cos square x is equal to -sin2x. In this post, we will find the derivative of cos square x. Derivative of cos^2 x by First Principle The first principle of derivatives says that the derivative of a function f(x) is … Read more
The derivative of sin^2x by first principle is equal to 2sinx cosx=sin2x. Sin square x derivative is denoted by d/dx (sin2x) and its formula is given as follows: $\dfrac{d}{dx}$(sin2x) = 2sinx cosx = sin2x. In this post, we will find the derivative of sine square x. Derivative of sin square x by First Principle The … Read more
The quotient rule of limit says that the limit of the quotient of two functions is the same as the quotient of the limit of the individual functions. In this post, we will prove the quotient law of limit by the epsilon-delta method. Question: What is an epsilon-delta proof of the division rule of limit? Here, … Read more