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(ex)/dx. An exponential function is of the form f(x) = ax, 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(ex)/dx = ex.
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.
1. | What is the Differentiation of e to the Power x? |
2. | Differentiation of e to the Power x Formula |
3. | Differentiation of e to the Power x Using First Principle |
4. | Differentiation of e to the Power x Using Derivative of ax |
5. | FAQs on Differentiation of e to the Power x |
What is the Differentiation of e to the Power x?
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 ex. Mathematically, it is denoted as d(ex)/dx = ex. 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) = ex, 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 ax.
Differentiation of e to the Power x Formula
Suppose y = ex ⇒ ln y = ln ex ⇒ ln y = x. On differentiating this with respect to x, we have (1/y) dy/dx = 1 ⇒ dy/dx = y ⇒ dy/dx = ex. If we differentiate e to the power x with respect to x, we have d(ex)/dx = ex. Hence the formula for the differentiation of e to the power x is,
Differentiation of e to the Power x Using First Principle of Derivatives
Next, we will prove that the differentiation of e to the power x is equal to ex 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) = limh→0 [f(x + h) - f(x)] / h
- ex + h = ex.eh
- lim x→0 (ex - 1) / x = 1
Using the above formulas, we have
d(ex)/dx = limh→0 [ex + h - ex] / h
= limh→0 [ex.eh - ex] / h
= limh→0 ex [eh - 1] / h
= ex limh→0[eh - 1] / h
= ex × 1
= ex
Hence we have proved the differentiation of e to the power x to be equal to e to the power x.
Differentiation of e to the Power x Using Derivative of ax
An exponential function is of the form f(x) = ax, where 'a' is a constant (real number) and x is the variable. The derivative of exponential function f(x) = ax is f'(x) = (ln a) ax. Using this formula and substituting the value a = e in f'(x) = (ln a) ax, we get the differentiation of e to the power x which is given by f'(x) = (ln e) ex = 1 × ex = ex [Because by log rules, ln e = 1]. Hence, the derivative of e to the power x is ex.
Important Notes on Differentiation of e to the Power x:
- The nth differentiation of e to the power x is equal to ex, that is, dn(ex)/dxn = ex
- The derivative of the exponential function with base e is equal to ex.
- The derivative of eax is aeax. Using this formula, we have the differentiation of ex to be 1.ex = ex.
☛Related Topics:
- Derivative of e2x
- Derivative of Exponential Function
- Derivative of log x
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 power x is equal to e to the power x itself because the derivative of an exponential function with base 'e' is equal to ex. Mathematically, it is denoted as d(ex)/dx = ex.
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 = -ex.
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(esin x)/dx = cos x esin 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(ex ln x)/dx = ex lnx (1 + lnx).This follows from chain rule.
How do you Find the Derivative of an Exponential Function?
The derivative of exponential function f(x) = ax is f'(x) = (ln a) ax 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) = ax is f'(x) = (ln a) ax. If a = e, then the formula for the differentiation of e to the power is ex.