Differentiation of e to the Power x

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

Let us learn about the differentiation of e to the power x and some variations of the function e to the power x. We will determine the differentiation of e to the power x using different methods including the first principle of differentiation, and derivative of exponential function along with some examples for a better understanding.

The differentiation of e to the power x is equal to e to the power x because the derivative of an exponential function with base 'e' is equal to e^x. Mathematically, it is denoted as d(e^x)/dx = e^x. e to the power x is an exponential function with a base equal to 'e', which is known as "Euler's number". It is written as f(x) = e^x, where 'e' is the Euler's number and its value is approximately 2.718. The differentiation of e to the power x can be done using different methods such as the first principle of differentiation and derivative of a^x.

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

Next, we will prove that the differentiation of e to the power x is equal to e^x using the first principle of differentiation. We know that for two exponential functions, if the bases are the same, then we add the powers. To prove the derivative of e to the power x, we will use the following formulas of exponential functions and derivatives:

• f'(x) = lim _h→0 [f(x + h) - f(x)] / h
• e^{x + h} = e^x.e^h
• lim _x→0 (e^x - 1) / x = 1

Using the above formulas, we have

d(e^x)/dx = lim _h→0 [e^{x + h} - e^x] / h

= lim _h→0 [e^x.e^h - e^x] / h

= lim _h→0 e^x [e^h - 1] / h

= e^x lim _h→0 [e^h - 1] / h

= e^x × 1

= e^x

Hence we have proved the differentiation of e to the power x to be equal to e to the power x.

An exponential function is of the form f(x) = a^x, where 'a' is a constant (real number) and x is the variable. The derivative of exponential function f(x) = a^x is f'(x) = (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 e to the power x which is given by f'(x) = (ln e) e^x = 1 × e^x = e^x [Because by log rules, ln e = 1]. Hence, the derivative of e to the power x is e^x.

Important Notes on Differentiation of e to the Power x:

• The n^th differentiation of e to the power x is equal to e^x, that is, dⁿ(e^x)/dxⁿ = e^x
• The derivative of the exponential function with base e is equal to e^x.
• The derivative of e^ax is ae^ax. Using this formula, we have the differentiation of e^x to be 1.e^x = e^x.

☛ Related Topics:

• Derivative of e^2x
• Derivative of Exponential Function
• Derivative of log x

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

The differentiation of e to the power x is equal to e to the power x itself because the derivative of an exponential function with base 'e' is equal to e^x. Mathematically, it is denoted as d(e^x)/dx = e^x.

What is the Differentiation of e to the Power 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 = -e^x.

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

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

What is the Derivative of e to the Power x log x?

The derivative of e to the power x log x is given by, d(e^{x ln x})/dx = e^{x ln x} (1 + ln x). This follows from chain rule.

How do you Find the Derivative of an Exponential Function?

The derivative of exponential function f(x) = 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) = a^x is f'(x) = (ln a) a^x. If a = e, then the formula for the differentiation of e to the power is e^x.