# Derivative e^x

Come friends today we will know about Derivative of e^x. In mathematics, the derivative of a function of a real variable measures the sensitivity of a change to a function value (the output value) with respect to a change in its argument (the input value). Derivatives are a fundamental tool of calculus. For example, the derivative of a moving object’s position with respect to time is the object’s velocity: it measures how quickly the object’s position changes as time progresses. The derivative of a function of a single variable at a chosen input value, when it is present, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function closest to that input value.

For this reason, the derivative is often described as the “instantaneous rate of change”, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of many real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph (after appropriate translation) is the best linear approximation to the graph of the original function. The Jacobian matrix is ​​the matrix that represents this linear transformation with respect to a given basis by the choice of independent and dependent variables. It can be calculated as the partial derivative with respect to the independent variable. For a real-valued function of many variables, the Jacobian matrix reduces to the gradient vector.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates differentiation with integration. Differentiation and integration constitute two fundamental operations in single-variable calculus. Let’s start with Derivative of e^x

## Definition of Derivative – Derivative e^x

A function of a real variable y = f(x) is differentiable at a point a of its domain, if its domain contains an open interval I containing a, and the limit

{\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}

exists. This means that, for every positive real number ϵ (even very small), there exists a positive real number δ such that, for every h such that | h | < δ and

{\displaystyle h\neq 0}~~ then ~~{\displaystyle f(a+h)}{\displaystyle f(a+h)}~~ is

defined, and

{\displaystyle \left|L-{\frac {f(a+h)-f(a)}{h}}\right|<\varepsilon ,}


where the vertical bars denote the absolute value (see (ε, δ)-definition of limit).

If the function f is differentiable at a, that is if the limit L exists, then this limit is called the derivative of f at a, and denoted f (a) (read as “f prime of a“) or df/dy = (a) (read as “the derivative of f with respect to x at a”, “dy by dx at a”, or “dy over dx at a”); see § Notation (details), below.

## Definition of e

In mathematics, e is an empirical number. Its value is approximately 2.71828. It is also sometimes called ‘Euler’s number’. e is an important mathematical constant. This number is taken as the base of the natural logarithm.

e is defined by the following two expressions-

{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}
{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =1+1+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+\cdots }

## Derivative e^x

#### Proof of

\frac{d}{dx}~(e^x)~~~~~~;~~~~~~~~~~~by~~~~~~~~~~~\frac{d}{dx}ln(x)

Given ;

\frac{d}{dx}~=~ln(x)~=~\frac{1}{x}~~~~;~~~~~Chain~~ Rule;~~~~~~\frac{d}{dx}x=1

Solve

\frac{dy}{dx}~=~ln(e^x)~\frac{d}{dx}x=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....(1)
\frac{d}{dx}~ln(e^x)=\frac{d}{dx}ln(u)\frac{d}{dx}e^x~~~~~~~~~(Let's~~~u~~=~~e^x)
\frac{1}{u}~\frac{d}{dx}(e^x)=\frac{1}{e^x}\frac{d}{dx}e^x~=~1~~~~~~~~~~~~~~~~~~~(equation ~1)
\frac{d}{dx}e^x=e^x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.............Answer`

2

#### What is the derivative of 4x?

The derivative of 4 4x is 4 .

#### What is the derivative of 2t?

2 is a constant whose value never changes. Thus, the derivative of any constant, such as 2 , is,

#### What is the derivative of 1 x 2?

Explanation: We will use the power rule, which states that the derivative of xn is nxn−1. We can use the power rule to write 1×2 as x−2. Thus, according to the power rule, the derivative of x−2 is −2x−2−1=−2x−3=−2×3 .

#### Why is the derivative of x2 2x?

Since f(x) = x², ‘x’ on the x-axis results in x’ on the y-axis. Similarly, x+δ on the x-axis results in a (x+δ)² on the y-axis. … Then we simplify the question, which results in 2x. We have now proved that the difference of x² is equal to 2x.

#### What is the difference of 0?

The derivative of 0 is 0 . In general, to find the derivative of a stationary function, we have the following rule, f(x) = a.

#### How do you find maxima and minima?

Answer: Finding the relative maxima and minima for a function can be done by looking at the graph of that function . A relative maxima is a point greater than the points directly adjacent to either side. Whereas, a relative minimum is any point that is less than the points directly adjacent to it on either side.

#### What is the derived formula?

A derivative helps us to know the changing relationship between two variables. Mathematically, the derived formula is helpful for finding the slope of a line, for finding the slope of a curve, and for finding the change in one measurement with respect to another measure. The derived formula is ddx x n = n. xn−1 ddx .

#### What does derivative mean in mathematics?

The derivative, in mathematics, is the rate of change of a function with respect to a variable . … Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more accurately, as the slope of the tangent line at a point.

#### How do you differentiate FX?

Apply the power rule to isolate a function. The power law states that if f(x) = x^n or x is raised to the power n, then f'(x) = nx^(n – 1) or x is raised to the degree (n – 1) and multiplied by n. For example, if f(x) = 5x, then f'(x) = 5x^(1 – 1) = 5.

#### What is the derivative of cosine?

-sin x

The derivative of the cosine function is written as (cos x)’ = -sin x, that is, the derivative of cos x is -sin x.See also How to magnetize something

#### Is pi constant in derivatives?

The derivative of is 0. The number is an irrational number whose approximate value is 3.14. Therefore, is a constant .

#### What is maximum and minimum in maths?

In mathematics, the maximum and minimum of the set A is A. The largest and smallest element of . They are written as else. , respectively. Similarly, the maximum and minimum values ​​of a function are the largest and smallest values ​​that the function takes at a given point.