how to find the derivative of e

Differentiation of due east to the Power x

Differentiation of due east to the power x is a process of determining the derivative of e to the power ten with respect to x which is mathematically written as d(e¹⁰)/dx. An exponential function is of the form f(10) = a^x, where 'a' is a real number and x is a variable. e to the power ten is an exponential function with base of operations (a) equal to the Euler's number 'east' and the differentiation of due east to the power x is equal to e to the power x, that is, itself. It is written as d(east^x)/dx = e¹⁰.

Let us learn well-nigh the differentiation of eastward to the power 10 and some variations of the part e to the power x. We will determine the differentiation of e to the ability 10 using different methods including the showtime principle of differentiation, and derivative of exponential role along with some examples for a better understanding.

1.	What is the Differentiation of e to the Power x?
ii.	Differentiation of e to the Ability x Formula
3.	Differentiation of e to the Ability x Using First Principle
4.	Differentiation of e to the Power 10 Using Derivative of a10
5.	FAQs on Differentiation of due east to the Power x

What is the Differentiation of due east to the Power ten?

The differentiation of e to the ability ten is equal to e to the power x because the derivative of an exponential function with base 'e' is equal to eastward^x. Mathematically, it is denoted every bit d(e^x)/dx = e^ten. e to the ability x is an exponential function with a base equal to 'e'. Information technology is written as f(10) = e^ten, where 'e' is the Euler's number and its value is approximately 2.718. The differentiation of eastward to the ability x can be done using different methods such as the get-go principle of differentiation and derivative of a^ten.

Differentiation of e to the Power ten Formula

Suppose y = due east^ten ⇒ ln y = ln e^x ⇒ ln y = x. On differentiating this with respect to x, nosotros have (1/y) dy/dx = i ⇒ dy/dx = y ⇒ dy/dx = due east^x. If we differentiate due east to the power x with respect to x, nosotros have d(e^x)/dx = eastward^x. Hence the formula for the differentiation of east to the power ten is,

Differentiation of e to the Power ten Using First Principle of Derivatives

Next, nosotros volition testify that the differentiation of e to the power x is equal to due east^x using the offset principle of differentiation. We know that for two exponential functions, if the bases are the same, then we add together the powers. To prove the derivative of e to the ability 10, we volition utilise the following formulas of exponential functions and derivatives:

• f'(x) = lim_h→0 [f(x + h) - f(x)] / h
• e^{x + h} = eastward^x.e^h
• lim _x→0 (e^x - one) / x = ane

Using the to a higher place formulas, nosotros have

d(e^x)/dx = lim_h→0 [e^{10 + h} - due east^x] / h

= lim_h→0 [e^x.eastward^h - e¹⁰] / h

= lim_h→0 e^x [eastward^h - 1] / h

= due east^x lim_h→0 [e^h - 1] / h

= e^x × 1

= e^x

Hence we take proved the differentiation of e to the power x to be equal to due east to the power x.

Differentiation of e to the Ability x Using Derivative of a^x

An exponential function is of the form f(x) = a^ten, where 'a' is a constant (real number) and x is the variable. The derivative of exponential role f(x) = a^x is f'(ten) = ln a a^x. Using this formula and substituting the value a = e in f'(x) = ln a a^x, we get the differentiation of due east to the power ten which is given by f'(x) = ln due east e^x = i × eastward^x = e^ten [Because ln e = 1]. Hence, the derivative of eastward to the power 10 is due east^x.

Important Notes on Differentiation of due east to the Power ten

• The n^thursday differentiation of east to the power x is equal to eastward^x, that is, d^north(e^x)/dxⁿ = e^x
• The derivative of the exponential function with base e is equal to eastward^x.
• The derivative of due east^ax is ae^ax. Using this formula, we accept the differentiation of e^ten to be 1.east^x = eastward^x.

Related Topics

• Derivative of e^2x
• Derivative of Exponential Office
• Derivative of log ten

Differentiation of e to the Ability x Examples

1. Example i: What is the differentiation of e to the power x to the power 2?

   Solution: To make up one's mind the differentiation of e to the power ten to the power ii, that is, \(e^{x^{2}}\), we volition use the chain dominion.

   We know that for y = f(grand(x)), the derivative is y' = f'(g(x)) × grand'(x). Let u = g(x) = x² and y = f(g(x)) = \(e^{ten^{2}}\) ⇒ y = f(u) = e^u

   We have dy/dx = dy/du × du/dx

   = east^u × 2x

   = \(2xe^{10^{2}}\)

   Answer: Hence the derivative of \(due east^{x^{2}}\) is \(2x eastward^{x^{2}}\)

2. Example two: Decide the differentiation of e to the power x sin 10.

   Solution: To evaluate the value of the derivative of e^xsinx, we will use production rule of differentiation.

   d(eastward^xsinx)/dx = (e^x)' sin x + (sin x)' due east^x

   = e^x sin 10 + e^ten cos x [Because derivative of sin ten is cos x]

   = e^x (sin x + cos 10)

   Answer: Hence differentiation of e to the ability x sin x is e¹⁰ (sin ten + cos x).

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Differentiation of eastward to the Ability x Questions

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FAQs on Differentiation of e to the Power x

What is the Differentiation of e to the Power x in Calculus?

The differentiation of e to the ability 10 is equal to e to the power 10 because the derivative of an exponential function with base of operations 'due east' is equal to east^ten. Mathematically, information technology is denoted as d(e^ten)/dx = e^x.

What is the Differentiation of e to the Ability Minus x?

The differentiation of e to the power minus x is equal to the negative of e to the power minus x, that is, d(e^{- x})/dx = -east^ten.

What is the Differentiation of e to the Power Sin x?

The differentiation of eastward to the power sin ten is equal to the production of cos x and due east to the power sin x, that is, d(e^{sin x})/dx = cos x e^{sin ten}.

What is the Derivative of due east to the Ability x log x?

The derivative of e to the power x log x is given by, d(e^{10 log ten})/dx = eastward^{x log x} (1 + log 10).

How practise you Find the Derivative of an Exponential Part?

The derivative of exponential part f(10) = a^x is f'(x) = ln a a^x which can be calculated by using the first principle of differentiation.

What is the Formula for Exponential Differentiation?

The formula for exponential differentiation for f(x) = a¹⁰ is f'(ten) = ln a a^x. If a = eastward, and then the formula for the differentiation of e to the power is e¹⁰.

Source: https://www.cuemath.com/calculus/differentiation-of-e-to-the-power-x/

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