## Derivative of Sin Inverse x: Formula, Proof [First Principle]

Derivative of sine inverse x. In this article, we will find the derivative of sin inverse x using the substitution method and the limit definition. In the end, we will evaluate the derivative of sine inverse 1/x as an application. What is the Derivative of Sin Inverse x? (Substitution Method) Step 1: Let y=sin-1 x … Read more

## Derivative of e^2x: Formula, Proof by First Principle, Chain Rule

What is the Derivative of e2x? The derivative of e2x is 2e2x. Here, we will find the derivative of e2x using the derivative of ex. We will use three methods: substitution method, chain rule, and limit definition. Derivative of e2x Formula The derivative of e2x is 2e2x. This can be written mathematically as follows: d/dx(e2x) = 2 e2x  … Read more

## Derivative of Cube Root of x | Cube Root Derivative

Derivative of x1/3. In this article, we will find the derivative of the cube root of x by applying the power rule of derivatives. We will also use the limit definition to evaluate the derivative of the cube root of x. What is the derivative of cube root of x? First Method of Finding Derivative of … Read more

## Derivative of Square Root of x | Root x Derivative

Derivative of root x. The square root of x is an important function in mathematics. So it is natural to study the derivative of the square root of x. We will use the formula of power rule of derivatives to find it. We will also evaluate the derivative of the square root of x by … Read more

## Derivatives of Inverse Trigonometric Functions

In this section, we will learn the derivative formulas of the inverse trigonometric functions with their proofs. Now we calculate the derivative of $\sin^{-1}x$ (or arc sin x). Derivative of $\sin^{-1}x$ $\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}} \, (|x|<1)$ Proof: Note that $\sin^{-1}x$ is defined when $|x| \leq 1.$ We assume that $y=\sin^{-1}x.$ As $|x| \leq 1$, we must have … Read more

## Proofs of Derivative Formulas

Here we will calculate the derivatives of some well-known functions from the first principle. For example, we will find the derivatives of the polynomial functions, trigonometric functions, exponential functions, logarithmic functions, and so on. Firstly, we find the derivative of xn using the definition of the derivative. Power rule of Derivative using First Principle: $\frac{d}{dx}(x^n)=nx^{n-1}$ … Read more

## Proofs of Derivative Properties with Examples

Here we will prove various properties of derivatives with applications one by one. Derivative of a constant function is zero- proof: For a constant $c$, we have $\frac{d}{dx}(c)=0$ Proof: Let $f(x)=c$ Now, $\frac{d}{dx}(c)$ $=\frac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0}{ \large \frac{f(x+h)-f(x)}{h} }$ $=\lim\limits_{h \to 0}{ \large \frac{c-c}{h} }$ $=\lim\limits_{h \to 0}{ \large \frac{0}{h} }$ $=\lim\limits_{h \to 0}0$ … Read more

## Derivative: definition, formulas, properties, and examples

The concept of the derivative is the backbone of the theory of Calculus. Here we will learn the definition of the derivative of a function, its various formulas, and its properties with examples.   Definition of the Derivative of a function: Limit definition of derivative: Let f(x) be a function of the variable x. Then … Read more

## Derivative of a Function : Definition, Properties, Examples

The theory of 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 $x$ be a … Read more