Derivative of x^4 by First Principle, Power Rule

The derivative of x^4 is equal to 4×3. In this post, we will learn how to differentiate x to the power 4 by the power rule and the first principle of derivatives. The formula of the derivative of x4 is given below. $\dfrac{d}{dx}(x^4)=4x^3$. Derivative of x^4 by Power Rule Recall the power rule of derivatives: … Read more

Derivative of a^x by First Principle

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

Derivative of 1/(1+x) by First Principle

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

Derivative of Root logx | Root logx Derivative

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

Derivative of 1/(1+x): Proof by Chain, Quotient Rule

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

Derivative of (ax+b)/(cx+d) | (ax+b)/(cx+d) Derivative

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 mod x | Mod x Derivative

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

Derivative of x^x [x to the power x] | d/dx(x^x)

The derivative of xx is equal to xx(1+ln x). That is, d/dx(xx)=xx(1+ln x). This can be proved by the logarithmic differentiation. Here, ln=loge. In this post, we will learn how to find the derivative of x to the x. Derivative of xx Formula The formula of xx derivative is given by d/dx(xx)=xx(1+ln x) Derivative of … Read more

Derivative of cos2x by First Principle, Chain Rule

The derivative of cos2x is equal to -2sin2x.  In this post, we will find the derivative of cos2x by the first principle, that is, by the limit definition of derivatives as well as by the chain rule of derivatives. Recall the first principle of derivatives. By this rule, we know that the derivative of a … Read more