promath In this section, we will prove that the absolute value of x is not differentiable at the point x=0. In other words, the function |x| is not differentiable at x=0. Absolute Value of x is not Differentiable at 0 The function f(x)=|x| is defined as follows: $|x|=\begin{cases} x, & \text{ if } x\geq 0 \\ … Read more
Derivative implies Continuity but Converse NOT True: In this section, we will prove that if a function is differentiable at a point, then the function is continuous at that point. Its converse statement is the following: if a function is continuous, then it is not necessarily differentiable. We prove the converse statement by providing examples. … Read more
Concept of Derivative/Differentiation: The theory of Derivative/Differentiation is the backbone of Calculus. With the help of differentiation, we actually determine the rate of changes of the dependent variable with respect to the independent variable. In this section, we will discuss the concept of derivatives. Here we go. 👩 Few Definitions: The increment of a variable: Let … Read more
Introduction to Integral Calculus: In Differential Calculus, we have learned how to find the derivative/differential of a differentiable function. The study of the inverse method of differential calculus is the main purpose of integral Calculus. We now understand this with an example. Let f(x) be a function with $f'(x)=\frac{d}{dx}$(f(x)) = 3×2. From this information, can … Read more
Properties of Derivatives and Their Proofs In this section, we will prove various properties of derivatives with applications. The following notation will be used to understand the derivative of $f(x)$: $f'(x)=\dfrac{d}{dx}(f(x))$.  Theorem 1: Prove that the derivative of a constant function is zero. Proof: Let $f(x)=c$ be a constant function. We have to show that … Read more
Proofs of Derivative Formulas by Definition: In this section, we will evaluate derivatives of functions u\sing the first principle. For example, we calculate the derivatives of $x^n$, $e^x$, $\sin x$, $\log x$ etc. The following formulas will be proved by the definition of derivatives: 1. $\frac{d}{dx}$(xn)=nxn-1 2. $\frac{d}{dx}$(ex)=ex 3. $\frac{d}{dx}$(ax) = ax logea 4. $\frac{d}{dx}$(log … Read more
In this blog post, we will list all the important formulas of derivatives along with its properties. The problems related to differential calculus can be easily solved if you have a complete list of derivative/differential formulas in your table. So we provide here a complete list of basic derivative formulas to help you. Definition of … Read more
Integration by Substitution Method: Some Important Forms with Examples Let f(x) be an integrable function in one variable x. If we have to integrate some well-known twisted form of f(x), then we may take the help of the substitution method. Here we list some important forms. Form 1: $\int f(ax+b) dx$ can be solved by the … Read more
Trigonometric Limits: Problems and Solutions We will compute the limits of trigonometric functions. A few examples of trigonometric functions are `sin x, cos x, tan x, sin (x+a), frac{sin x}{x}` etc. At first, recall the well-known formulas of trigonometric limits. 1. `lim_{x to 0} sin x =0` 2. `lim_{x to 0} cos x =1` 3. `lim_{x to 0} … Read more
Problems and Solutions of Basic Limits: Here we will discuss various basic problems of limits with solutions. Example 1: Evaluate `lim_{x to 0} (sin x + cos x)` Solution: `lim_{x to 0} (sin x +cos x)` `=lim_{x to 0} sin x + lim_{x to 0}cos x` `=sin 0 + cos 0` `=0+1` `=1.` Example 2: Evaluate … Read more