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.
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 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}.
Differentiation of e to the Power x Formula
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,
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 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.
Differentiation of e to the Power x Using Derivative of a^{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, d^{n}(e^{x})/dx^{n} = 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}.
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Differentiation of e to the Power x Examples

Example 1: What is the differentiation of e to the power x to the power 2?
Solution: To determine the differentiation of e to the power x to the power 2, that is, \(e^{x^{2}}\), we will use the chain rule.
We know that for y = f(g(x)), the derivative is y' = f'(g(x)) × g'(x). Let u = g(x) = x^{2} and f(x) = e^{x} and y = f(g(x)) = \(e^{x^{2}}\) ⇒ y = f(u) = e^{u}
We have dy/dx = dy/du × du/dx
= e^{u} × 2x
= \(2xe^{x^{2}}\)
Answer: Hence the derivative of \(e^{x^{2}}\) is \(2x e^{x^{2}}\)

Example 2: Determine the differentiation of e to the power x sin x.
Solution: To evaluate the value of the derivative of e^{x}sinx, we will use product rule of differentiation.
d(e^{x}sinx)/dx = (e^{x})' sin x + (sin x)' e^{x}
= e^{x} sin x + e^{x }cos x [Because derivative of sin x is cos x]
= e^{x} (sin x + cos x)
Answer: Hence differentiation of e to the power x sin x is e^{x} (sin x + cos 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 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}.
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