## Derivative of mod x: Formula, Proof

The derivative of |x| (mod x) is equal to x/|x| where x is non-zero. The function |x| is known as the absolute value of x. So the derivative of absolute value of x is given by $\dfrac{d}{dx}(|x|) = \dfrac{x}{|x|}$, provided that x ≠ 0. Before we prove this formula, let us first recall the definition … Read more

## Derivative of x root(x): by First Principle, Power Rule

The derivative x root(x) is equal to (3√x)/2. Note x root(x) is denoted by x√x, and its derivative as d/dx (x√x). In this post, let us learn how to differentiate x root(x) using the following methods: The derivative formula of x root(x) is given as follows. $\dfrac{d}{dx}(x\sqrt{x}) = \dfrac{3\sqrt{x}}{2}.$ By Power Rule To find the … Read more

## Derivative of Fourth Root of x: Proof by First Principle

The derivative of fourth foot of x is equal to 1/(4×3/4). Fourth root of x is denoted by ∜x = x1/4, so its derivative formula is given by $\dfrac{d}{dx}$(∜x) = $\dfrac{1}{4x^{3/4}}.$ In this article, we will learn how to differentiate fourth root of x with respect to x by the following methods: By Power Rule … Read more

## Chain Rule of Derivatives: Statement, Formula, Proof, Examples

The chain rule of derivatives is used to find the derivative of a composite function. The rule states that the derivative of a composite function f(g(x)) is equal to f'(g(x)) ⋅ g'(x). In this post, we will learn the statement of the chain rule, its proof, step-by-step method to use this rule along with solve … Read more

## Derivative of sec x: Formula, Proof by First Principle, Chain, Quotient Rule

The derivative of sec x with respect to x is equal to secx tanx. The secx is the reciprocal of cosx. In this post, we will learn how to find the derivatives of sec x using the following methods: What is the Derivative of Sec x? The derivative of secx with respect to x is … Read more

## Derivative of cotx: Proof by First Principle, Product, Quotient, Chain Rule

The derivative (or differentiation) of cot x is equal to -cosec2 x. Let’s learn how to find the derivative of cot x using the following method of derivatives: What is the Derivative of cot x? The derivative of cot x is denoted by the symbol $\frac{d}{dx}$(cot x) or (cot x)$’$ and it is equal to … Read more

## Derivative of Sin Square x | Sin^2x Derivative

The derivative of sin square x is equal to 2sinx cosx (or sin2x). Note that sin2x is the square of sinx. In this article, we will find the derivative of sin2x by the following methods: Derivative of sin2x by Product Rule Step 1: At first, we write sin2x as a product of two copies of … Read more

## Derivative of x^x: Formula, Proof by First Principle

The derivative of xx (x to the power x) is equal to xx(1+logex). In this post, we will learn the formula for the derivative of xx and how to find it. To calculate the derivative of x to the x, we will use the following methods: Derivative of xx Formula The derivative of xx is … Read more

## Calculus 1 Final Exam Review

In this article, we will review the final exam of Calculus 1 in detail and chapterwise. The syllabus of Calculus 1 includes Limit, Continuity, Derivative, Integration, and their applications. Limit Final Exam Review Question 1: Find $\lim\limits_{x \to 2} \dfrac{x^2-5x+6}{x^2-4}$. Solution: At first, we will factorize both the numerator and the denominator. Note that x2-5x+6 … Read more

## Fixed Point Theorem: Statement, Proof, Examples

As an application (or an example) of the intermediate value theorem, we can prove the fixed point theorem (FPT) for continuous function which is given below. Let us first recall the intermediate value theorem. Intermediate Value Theorem: If f(x) is a real-valued continuous function on the closed interval [a, b] with f(a) ≠ f(b), then … Read more