The integral of e^(-x^2) from 0 to infinity is equal to √π/2, that is, ∫exp(-x^2) dx = √π/2. Here we will learn how to integrate e-x^2 (e to the power -x2) from 0 to ∞. Integration of exp(-x^2) from 0 to infinity Answer: The integral of $e^{-x^2}$ from 0 to infinity is equal to √π/2. … Read more
The derivative of ln(ln(x)) is equal to 1/{x lnx ln(lnx)}. This is obtained by using the chain rule of differentiation. In this post, lets learn how to differentiate ln(ln(lnx)). Differentiation of ln(ln(lnx)) Step 1: Recall, the chain rule of differentiation: Suppose $f$ is a function of $z$ and $z$ is a function of $x$, that … Read more
No, a continuous function is not always differentiable. For example, f(x)=|x| is continuous but not differentiable at x=0. In this post, we will study the following: is a continuous function always differentiable? A continuous function may not be differentiable always. Although, a differentiable function is always continuous. Lets prove this fact here. Recall the definitions … Read more
The nth derivative of cosx is equal to cos(nπ/2 +x). The nth derivative of cos x is denoted by dn/dxn (cosx), and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( \cos x\right)=\cos \left(\dfrac{n \pi}{2}+x \right)}$ nth Derivative of cos x Question: Find the nth derivative of cosx. Answer: To find the nth derivative of cosx with … Read more
The nth derivative of sinx is equal to sin(nπ/2 +x). The nth derivative of sin x is denoted by dn/dxn (sinx), and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( \sin x\right)=\sin \left(\dfrac{n \pi}{2}+x \right)}$ nth Derivative of sin x Question: Find the nth Derivative of sinx. Answer: To find the nth derivative of sinx with … Read more
The nth derivative of 1/(ax+b) is equal to (-1)nn!an/(ax+b)n+1. The nth derivative of 1/(ax+b) is denoted by $\dfrac{d^n}{dx^n}\left( \dfrac{1}{ax+b}\right)$ and its formula is given below: $\boxed{\dfrac{d^n}{dx^n}\left( \dfrac{1}{ax+b}\right)=\dfrac{(-1)^n n! a^n}{(ax+b)^{n+1}}}$ nth Derivative of 1/(ax+b) Question: What is the nth Derivative of $\dfrac{1}{ax+b}$? Answer: Let us put y = $\dfrac{1}{x+b}$ = (ax+b)-1. Using the power rule $\dfrac{d}{dx}\left( … Read more
The nth derivative of xn is equal to n!. The nth derivative of x^n is denoted by $\frac{d^n}{dx^n}\left( x^n\right)$, and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( x^n\right)=n!}$ nth Derivative of xn Question: Find nth Derivative of xn. Answer: The nth derivative of x to the power n is obtained by repeatedly using the power … Read more
The partial derivative of log(x^2+y^2) with respect to x is equal to 2x/(x2+y2) and with respect to y is equal to 2y/(x2+y2). So their formulas are as follows: Function Partial Derivative z=log(x2+y2) ∂z/∂x = 2x/(x2+y2) z=log(x2+y2) ∂z/∂y = 2y/(x2+y2) where ∂z/∂x is the partial derivative of z with respect to x. Partial Derivative of log(x2+y2) … Read more
If y=cos(x+y), then dy/dx= -sin(x+y)/[1+sin(x+y)]. Here, we learn how to differentiate y=cos(x+y) with respect to x. Let us find the derivative of y=cos(x+y). Find dy/dx if y=cos(x+y) Question: If y=cos(x+y), then $\dfrac{dy}{dx}$. Solution: We are given that y = cos(x+y). Step 1: Differentiating both sides of the above equation with respect x, we get that … Read more
If y=sin(x+y), then dy/dx= cos(x+y)/[1-cos(x+y)]. Here, we learn how to differentiate y=sin(x+y) with respect to x. Let us find the derivative of y=sin(x+y). y=sin(x+y), Find dy/dx Question: If y=sin(x+y), then $\dfrac{dy}{dx}$. Solution: Given, y = sin(x+y). To find dy/dx, we will differentiate both sides of the equation y=sin(x+y) with respect x. Using the chain rule, … Read more