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
The product rule of limits says that the limit of the product of two functions is the same as the product of the limits of the individual functions. In this post, we will prove the product law of limits by the epsilon-delta method. Question: What is an epsilon-delta proof of the product law of limits? … Read more
The sum rule of limits says that the limit of the sum of two functions is the same as the sum of the limits of the individual functions. In this post, we will prove the sum/addition rule of limits by the epsilon-delta method. Question: What is an epsilon-delta proof of the sum law of … Read more
The cube root of x is a number whose cube is x, in other words, (cube root of x)^3 = x. In this post, we will evaluate the integration of the cube root of x. To do so, we will use the power rule of derivatives which is given below: $\displaystyle \int x^n dx = … Read more
The derivative of root x minus 1 is 1 by 2 root x. In this post, we will prove this derivative formula by the power rule of derivatives as well as by the first principle of derivatives. Derivative of root(x)-1 Formula The Formula for the derivative of root x -1 is given below: Derivative of … Read more