Derivative of e to the Power x (ex)
We know that the derivative of exponential function ax is given as ax ln a where a is a real number and greater than 0.
Now if we assume, a = e in ax as e is a real number and its value is 2.718… i.e. greater than zero then our function will be f(x) = ex
Now we have,
f(x) = ex
⇒ f'(x) = ex lne
We know that ln is a natural logarithm with base e i.e. logee
Now from logarithm formulas, we know that logaa = 1
Hence, ln e = logee = 1
Putting this value in f'(x) we get
f'(x) = ex ln e = ex
Derivative of e2x
We have f(x) = e2x which is an exponential function as it is in the form of ex but it is also a composite function consisting of two functions e2x and 2x. Hence, we will find the derivative of e2x using the chain rule.
Using the chain rule we know that the derivative of a composite function is given in the form of the derivative of the first function multiplied by the derivative of the second function.
f(x) = e2x ⇒ f'(x) = 2.e2x
Hence, the derivative of e2x is 2.e2x where e2x is derivative of e2x and 2 is derivative of 2x.
Derivative of Exponential Functions
Derivative of Exponential Function stands for differentiating functions which are expressed in the form of exponents. We know that exponential functions exist in two forms, ax where a is a real number r and is greater than 0 and the other form is ex where e is Euler’s Number and the value of e is 2.718 . . . On differentiating ax, we will get ax ln a and on differentiating ex, we will get ex.
Mathematically, the derivative of an exponential function is expressed as [Tex]\frac{d(ax)}{dx} = (ax)’ = ax \ln a [/Tex]. This derivative can be obtained through the first principles of differentiation, utilizing limit formulas. The graph of the derivative of an exponential function alters its direction when a > 1 and when a < 1.
In this article, we will learn about the derivative of the exponential function, its formula, proof of the formula, and examples in detail. But before learning about the differentiation of exponential function we must know about exponential function.
Table of Content
- Exponential Function Definition
- What is the Derivative of Exponential Function?
- Derivative of Exponential Function Formula
- Derivative of Exponential Function Proof
- Derivative of e to the Power x (ex)
- Exponential Function Derivative Graph
- Derivative of Exponential and Logarithmic Functions
- Derivative of Exponential Function Examples
- Derivative of Exponential Functions Worksheet