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 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

Discontinuity of a Function: Definition, Types, Examples

A function is discontinuous if we cannot sketch its graph without lifting the pen. In this article, we will study discontinuous functions with their types, examples, and a few solved problems. At first, we recall the definition of the continuity of a function. A function f(x) is called continuous at x=a if we have limx→af(x) … Read more

Continuity of Functions: Definition, Solved Examples

By continuity of a function, we mean that we can sketch the graph of the function without lifting the pencil. In this article, we will learn the definition of the continuity of a function along with its properties, examples, and solved problems. Definition of Continuity of a Function Let f(x) be a real-valued function where … Read more

nth Derivative: Definition, Formula, Properties, Examples

The nth derivative of a function is obtained by the successive differentiation of the same function till n times. n-th differentiation is referred to the higher order derivatives. In this article, we will learn the definition of the nth derivative along with its formulas, properties, and examples. nth Derivative Definition Let f(x) be a differentiable … Read more