promath 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
The integral of lnx is equal to ∫ln(x)= xln x -x+C where C is an integral constant. Here we will learn how to integrate lnx, that is, find ∫ln(x). The integral formula of lnx is given below: ∫ln(x) dx = xln(x) -x+C where ln(x) = logex. Integration of lnx To find the integration of lnx, … Read more
The Derivative of (ax+b)/(cx+d) is equal to (ad-bc)/(cx+d)2. In this post, we will learn how to differentiate the quotient function (ax+b)/(cx+d). The derivative of (ax+b)/(cx+d) is denoted by the symbol $\dfrac{d}{dx}\left( \dfrac{ax+b}{cx+d}\right)$ and it is equal to $\dfrac{d}{dx}\left( \dfrac{ax+b}{cx+d}\right)$ $=\dfrac{ad-bc}{(cx+d)^2}$ when the denominator cx+d is nonzero, that is, x ≠ -d/c. How to Differentiate (ax+b)/(cx+d) … Read more
Derivative of absolute value of x. The derivative of mod x is denoted by d/dx(|x|) and it is equal to x/|x| for all nonzero values of x. In this post, we will learn how to differentiate modulus x. Recall that mod x is defined as below. $|x|=\begin{cases} x, & \text{ if } x\geq 0 \\ … Read more
The integration of (ax+b)/(cx+d) is equal to ax/c + (bc-ad)/c2 ln|cx+d|+K where K is an integral constant and ln denotes the natural logarithm. In this post, we will learn how to integrate the function (ax+b)/(cx+d). Integration of (ax+b)/(cx+d) Formula The integration of (ax+b)/(cx+d) formula is given below. $\int \dfrac{ax+b}{cx+d} \ dx$ $=\dfrac{ax}{c} + \dfrac{bc-ad}{c^2} \ln|cx+d|+K$, … Read more
The integration of 1/x is equal to ln|x|+C where ln denotes the natural logarithm, that is, ln|x|=loge|x| and C is the integration constant. In this post, we will learn how to integrate 1/x. Integration of 1/x Formula The formula for the integration of 1/x (1 divided by x) is given below. ∫ 1/x dx = … Read more
The integration of 2x is equal to 2x/ln2 where ln denotes the natural logarithm, that is, ln=loge. In this post, we will learn to integrate 2 to the power x with respect to x. Integration of 2x Formula The integration formula of 2x (2 to the x) with respect to x is given below. ∫ … Read more