promath The integration of root x + 1/root x is equal to 2/3 x3/2+2√x+C where C is an integral constant. In this post, we will learn how to integrate √x+1/√x. Integral of √x+1/√x Question: Find the integral $\int (\sqrt{x}+\dfrac{1}{\sqrt{x}})\ dx$ Answer: $\int (\sqrt{x}+\dfrac{1}{\sqrt{x}})\ dx$ = $\int \sqrt{x}\ dx$ + $\int \dfrac{1}{\sqrt{x}}\ dx$ by the sum rule … Read more
The modulus of x, that is, |x| is a continuous function. In this post, we will learn how to prove mod x is continuous. Let us now show that mod x is continuous at the point x=0. f(x)=|x| is continuous at 0 Let f(x)=|x|. Note that we have So we obtain that $\lim\limits_{x \to 0} … Read more
The derivative of the modulus of sinx is equal to (sinx cosx)/|sinx|. In this post, we will learn how to differentiate mod sinx, that is, how to find d/dx(|sinx|). The differentiation formula of mod sinx is given below and we will learn how to derive this formula in details. Mod sinx Derivative Formula The derivative … Read more
The derivative of x^sinx (x to the power sinx) is equal to xsinx[sinx/x + cosx logx]. Here we learn how to differentiate x^sinx. Derivative of xsinx Formula xsinx Derivative Formula: The formula of xsinx derivative is given by d/dx(xsinx) = xsinx[sinx/x + cosx logx] Let us now give a proof of this fact. Derivative of … Read more
The derivative of 1/(cube root of x) is equal to $-\dfrac{1}{3\sqrt[3]{x^4}}$. Note that 1/(cube root of x) can be written mathematically as $\frac{1}{\sqrt[3]{x}}$. In this post, we will learn how to differentiate 1/(cube root of x). Derivative of 1/cube root x by Power Rule To find the derivative of 1/cube root of x, we will … Read more
The derivative of x^4 is equal to 4×3 which can by proved by first principle and power rule. The formula of the derivative of x4 is given below. $\dfrac{d}{dx}$(x4) = 4×3. Let us now learn how to differentiate x to the power 4 by the power rule and the first principle of derivatives. Derivative of … Read more
The derivative of a^x (a to the power x) is equal to axlna where ln denotes the natural logarithm, that is, lna=logea. In this post, we will learn how to differentiate a^x using the limit definition. The derivative formula of a^x is the following. $\dfrac{d}{dx}(a^x)=a^x\ln a$. The first principle or the limit definition of derivatives … Read more
The derivative of 1/(1+x) is equal to -1/(1+x)2. In this post, we will find the derivative of 1 divided by 1+x using the limit definition, that is, by the first principle. The first principle of derivatives says that the derivative of a function f(x) is given by the following limit: $\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ … Read more
The derivative of square root of logx is equal to 1/(2x root(logx)). In this post, we will learn how to differentiate root logx by the chain rule of derivatives. The formula for the derivative of root logx is given below. $\dfrac{d}{dx}(\sqrt{\log x})$ $=\dfrac{1}{2x\sqrt{\log x}}$ Derivative of Root logx by Chain Rule By chain rule of … Read more
The derivative of 1/(1+x) is equal to -1/(1+x)2. The function 1/(1+x) is the reciprocal of 1+x. In this post, we will learn how to differentiate 1 divided by 1+x. Its derivative is denoted by $\dfrac{d}{dx} \Big(\dfrac{1}{1+x} \Big)$ and it is equal to $\dfrac{d}{dx} \Big(\dfrac{1}{1+x} \Big)$ $=\dfrac{-1}{(1+x)^2}$. We will use the chain rule and the quotient … Read more