The derivative of $\tan x$ is $\sec^2 x$, that is, $\dfrac{d}{dx}(\tan x)=\sec^2 x$. In this post, we will learn how to find the derivative of \tan x by the product rule of derivatives as well as the quotient and chain rule of derivatives. At first, we will recall the rules. Let $f(x)$ and $g(x)$ … Read more
Here, we will find the derivative of root x+1/root x with respect to x. In the end, we will also evaluate this derivative at x=1. For more details of square roots, please click on the page Square Root of x: Definition, Symbol, Graph, Properties, Derivative, Integration Root(x)+1/Root(x) Derivative Question: Find the derivative of $\sqrt{x}+\frac{1}{\sqrt{x}}$ Answer: At … Read more
In this section, we will learn how to find To answer the question, let us first know the definition of the derivative. Definition of derivative: Let $f(x)$ be a differentiable function of $x$. From first principle of by definition, the derivative of $f(x)$ is given as follows: $f'(x)=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ Derivative of the Square … Read more
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
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
The following topics of Differential Calculus will be covered. Please click on the link provided below you wish to read. Derivative: Definition, Properties, and Formulas Proofs of Derivative Formulas (by definition) Some Interesting Problems & Solutions of Derivatives: 1. Derivative implies continuity but the converse is not true 2. Mod x is continuous at x=0 but not … Read more