Answer: The limit of x^x as x approaches 0+ is equal to 1. That is, limx→0+ xx = 1. Limit of xx when x tends to 0+ Question: Find the limit of xx when x tends to 0+? Solution We need to find limx→0+ xx. Step 1: Note that x = eln x ⇒ $x^x … Read more
Answer: The derivative of x^2 sin(1/x) at x=0 is equal to 0 if the function is defined to be 0 at x=0. Question Find the derivative of the function f(x) = $\begin{cases} x^2 \sin(\frac{1}{x}) & \text{if } x \neq 0 \\ 0 & \text{if } x=0 \end{cases}$ at x=0. That is, find f'(0). Solution From … Read more
The integral of e^(-x^2) from negative infinity to infinity is equal to √π, that is, $\int_{-\infty}^\infty$exp(-x^2) dx = √π. Here we will learn how to integrate e-x^2 (e to the power minus x2) from -∞ to ∞. Integral of exp(-x^2) from Minus Infinity to Infinity Answer: The integral of $e^{-x^2}$ from -∞ to ∞ is … Read more
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