Derivative of Exponential Function Proof
The derivative of the exponential function can be proved by using the first principle. In the First Principle Differentiation, we find the derivative of a function using the definition of limits. From the definition of limits, we know that
f'(x) = limh→0 f(x + h) – f(x) / h
Now we have function f(x) = ax
Now finding the derivative of f(x) = ax using the First Principle we have,
f'(x) = d(ax)/dx = limh→0 ax+h – ax / h
⇒ d(ax)/dx = limh→0 ax ⨯ ah – ax / h {From law of exponents we have, ax+h = ax ⨯ ah }
⇒ d(ax)/dx = limh→0 ax (ah – 1) / h
⇒ d(ax)/dx = ax limh→0 (ah – 1) / h
⇒ d(ax)/dx = ax ln a {Since, limh→0 (ah – 1) / h = ln a}
Hence, the derivative of the Exponential Function ax is the product of ax and the natural logarithm of a.
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