promath The limit of (a^x-1)/x as x approaches 0 is equal to 1. The limx -> 0 (ax-1)/x formula is given by $\lim\limits_{x \to 0} \dfrac{a^x-1}{x}=\ln a$ where ln a = logea. In this post, we will learn to prove the limit of (ax-1)/x when x tends to zero. Limx → 0 (ax -1)/x Prove that … Read more
The limit of log(1+x)/x as x approaches 0 is equal to 1. The limx -> 0 (log(1 + x))/x formula is given by $\lim\limits_{x \to 0} \dfrac{\log(1+x)}{x}=1$. In this post, we will learn to prove the limit of log(1+x)/x when x tends to zero. Limx → 0 log(1 + x)/x Prove that limx → 0 … Read more
The integral of 1/x^n (1 by x power n) is equal to x-n+1/(1-n) +C. Here we will learn how to integrate 1/xn using the power rule of integration. The integration formula of 1/xn is given by $\int \dfrac{dx}{x^n}=\dfrac{x^{1-n}}{1-n}+C$ where C denotes an integral constant. Integration of 1/xn The integral of 1/xn is denoted by ∫dx/xn. … Read more
The derivative of 1/xn (1 by x power n) is equal to -n/xn+1. Here we learn how to find the derivative of 1 divided by x to the n. The derivative formula of 1/x^n is given by $\dfrac{d}{dx} \big( \dfrac{1}{x^n}\big) = -\dfrac{n}{x^{n+1}}$. Derivative of 1/xn Answer: The derivative of 1/xn is equal to -n/xn+1. Explanation: … Read more
The integral of root cotx is equal to ∫√cotx dx = $-\dfrac{1}{\sqrt{2}}$ tan-1 ($\dfrac{\cot x-1}{\sqrt{2\cot x}}$) – $\dfrac{1}{2\sqrt{2}}$ log $|\dfrac{\cot x+1-\sqrt{2\cot x}}{\cot x+1+\sqrt{2\cot x}}|$ +C where C is an arbitrary integral constant. Here we will learn how to integrate square root of cotx. Integral of √cotx Question: Find ∫√cotx dx. Solution: Let cotx=t2. Differentiating w.r.t … Read more
The integration of root tanx + root cotx is equal to √2sin-1(sinx−cosx)+C. Here we learn how to integrate √tanx+√cotx. The integral formula of $\sqrt{\tan x}+\sqrt{\cot x}$ is given by $\int (\sqrt{\tan x}+\sqrt{\cot x}) dx$ = $\sqrt{2}\sin^{-1}(\sin x -\cos x)+C$. Find ∫(√tanx+√cotx) dx $\int (\sqrt{\tan x}+\sqrt{\cot x}) dx$ = $\int \Big(\dfrac{\sqrt{\sin x}}{\sqrt{\cos x}}+\dfrac{\sqrt{\cos x}}{\sqrt{\sin x}} \Big) … Read more
The derivative of ln(lnx) is equal to 1/xln(x) and it is denoted by d/dx (x lnx). So the derivative formula of ln(lnx) is given by $\dfrac{d}{dx}\big(\ln(\ln x) \big) = \dfrac{1}{x \ln x}$. Derivative of ln(lnx) by Chain Rule Answer: The derivative of ln(lnx) is $\dfrac{1}{x \ln x}$. Explanation: Let z=ln x. Differentiating, $\dfrac{dz}{dx}=\dfrac{1}{x}$. Now, $\dfrac{d}{dx}\big(\ln(\ln … Read more
The derivative of 4/x is equal to -4/x2. In this post, we will find the derivative of 4 divided by x by the power rule. The derivative of 4/x is denoted by d/dx(4/x), and its formula is given by $\dfrac{d}{dx}(\dfrac{4}{x})= – \dfrac{4}{x^2}$. Derivative of 4/x Question: Find the Derivative of 4/x. Answer: We can express … Read more
The derivative of 3/x is equal to -3/x2. The derivative formula of 3/x (3 divided by x) is given by as follows: $\dfrac{d}{dx}(\dfrac{3}{x})= – \dfrac{3}{x^2}$. Derivative of 3/x Question: What is the Derivative of 3/x? Answer: As 3/x can be expressed as 3x-1, applying the power rule, the derivative of 3/x is equal to -3/x². … Read more
The derivative of 2/x is equal to -2/x2. Here, we will learn how to find the derivative of 2 divided by x. The derivative formula of 2/x is given by $\dfrac{d}{dx}(\dfrac{2}{x})= – \dfrac{2}{x^2}$. Derivative of 2/x Question: Find the Derivative of 2/x. Answer: We can express 2/x as 2x-1. So by the power rule, the … Read more