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Derivative of Exponential Function
Before getting to the derivative of exponential function, let us recall the concept of an exponential function which is given by, f(x) = a^{x}, a > 0. One of the popular forms of the exponential function is f(x) = e^{x}, where 'e' is "Euler's number" and e = 2.718.... The derivative of exponential function f(x) = a^{x}, a > 0 is given by f'(x) = a^{x }ln a and the derivative of the exponential function f(x) = e^{x} is given by f'(x) = e^{x}.
In this article, we will study the concept of the derivative of the exponential function and its formula, proof, and graph along with some solved examples to understand better.
What is Derivative of Exponential Function?
The derivative of exponential function f(x) = a^{x}, a > 0 is the product of exponential function a^{x} and natural log of a, that is, f'(x) = a^{x} ln a. Mathematically, the derivative of exponential function is written as d(a^{x})/dx = (a^{x})' = a^{x} ln a. The derivative of exponential function can be derived using the first principle of differentiation using the formulas of limits. The graph of derivative of exponential function changes direction when a > 1 and when a < 1.
Derivative of Exponential Function e^{x}
Now that we know the derivative of the exponential function is given by f'(x) = a^{x} ln a, the derivative of exponential function e^{x} using the same formula is given by e^{x} ln e = e^{x} (because ln e = 1). Hence the derivative of exponential function e^{x} is the function itself, that is, if f(x) = e^{x}, then f'(x) = e^{x}.
Derivative of Exponential Function Formula
The formula for derivative of exponential function is given by,
 f(x) = a^{x}, f'(x) = a^{x} ln a or d(a^{x})/dx = a^{x} ln a
 f(x) = e^{x}, f'(x) = e^{x} or d(e^{x})/dx = e^{x}
Derivative of Exponential Function Proof
Now, we will prove that the derivative of exponential function a^{x} is a^{x} ln a using the first principle of differentiation, that is, the definition of limits. To derive the derivative of exponential function, we will some formulas such as:
 \(f'(x)=\lim_{h\rightarrow 0}\dfrac{f(x+h)f(x)}{h}\)
 \(\lim_{h\rightarrow 0}\dfrac{a^h1}{h} = \ln a\)
 a^{m} × a^{n }= a^{m+n}
Using the above formulas, we have
\(\begin{align} \frac{\mathrm{d} (a^x)}{\mathrm{d} x}&=\lim_{h\rightarrow 0}\dfrac{a^{x+h}a^x}{h}\\&=\lim_{h\rightarrow 0}\dfrac{a^x \times a^ha^x}{h}\\&=\lim_{h\rightarrow 0}\dfrac{a^x (a^h1)}{h}\\&=a^x\lim_{h\rightarrow 0}\dfrac{ a^h1}{h}\\&=a^x \ln a\end{align}\)
Hence we have derived the derivative of exponential function using the first principle of derivatives.
Graph of Derivative of Exponential Function
The graph of exponential function f(x) = b^{x} is increasing when b > 1 whereas f(x) = b^{x} is decreasing when b < 1. Thus, the graph of exponential function f(x) = b^{x}
 increases when b > 1
 decreases when 0 < b < 1
Since the graph of derivative of exponential function is the slope function of the tangent to the graph of the exponential function, therefore the graph of derivative of exponential function f(x) = b^{x }is increasing for b > 0. Given below is the graph of the exponential function and the graph of the derivative of exponential function for b > 1 and 1 < b < 1.
Important Notes on Derivative of Exponential Derivative
 Exponential function f(x) = a^{x} is not defined for a < 0 and so does the derivative of exponential function.
 d(a^{x})/dx = a^{x} ln a
 d(e^{x})/dx = e^{x}
Related Topics on Derivative of Exponential Function
Derivative of Exponential Function Examples

Example 1: Find the derivative of exponential function f(x) = 3^{x} + 3x^{2}
Solution: Using the formula for derivative of exponential function and other differentiation formulas, the derivative of f(x) = 3^{x} + 3x^{2} is given by,
f'(x) = 3^{x} ln 3 + 6x
Answer: The derivative of 3^{x} + 3x^{2} is 3^{x} ln 3 + 6x

Example 2: Differentiate the function f(x) = e^{x}/(1 + x) using the derivative of exponential function.
Solution: We will differentiate f(x) = e^{x}/(1 + x) by quotient rule. We know that the derivative of e^{x} is e^{x}
f'(x) = [e^{x}(1 + x)  e^{x}]/(1 + x)^{2}
= e^{x}(1 + x  1)/(1 + x)^{2}
= xe^{x}/(1 + x)^{2}
Answer: The derivative of f(x) = e^{x}/(1 + x) is xe^{x}/(1 + x)^{2}.
FAQs on Derivative of Exponential Function
What is the Derivative of Exponential Function?
The derivative of exponential function f(x) = a^{x}, a > 0 is the product of exponential function a^{x} and natural log of a, that is, f'(x) = a^{x} ln a.
What is the Derivative of Exponential Function e^{x}?
The derivative of exponential function e^{x} is the function itself, that is, if f(x) = e^{x}, then f'(x) = e^{x}.
Why is the Derivative of Exponential Function e^{x} itself?
The derivative of the exponential function a^{x} is given by f'(x) = a^{x} ln a. We know that ln e = 1 and if a = e, the derivative of exponential function e^{x} is given by e^{x} ln e = e^{x}
How to Find the nth Derivative of Exponential Function e^{x}?
Since the first derivative of exponential function e^{x} is e^{x}, therefore if we differentiate it further, the derivative will always be e^{x}. Hence the nth derivative of e^{x} is e^{x}.
Is the Derivative of Exponential Function e^{x }the Exponential Function?
Yes, the derivative of exponential function e^{x} is the exponential function e^{x} itself.
What is the General Derivative of Exponential Function?
The general derivative of exponential function a^{x} is given by a^{x} ln a.
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