The derivative of an exponential function, ax, is ax multiplied by the natural logarithm of a. Mathematically, this is represented as d(ax)/dx = (ax)' = ax ln a.
Understanding the Derivative
Here's a breakdown of what that means:
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Exponential Function: A function where the variable (x) is in the exponent. The general form is f(x) = ax, where a is a constant called the base.
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Derivative: The rate of change of a function with respect to a variable. In simpler terms, it tells you how much the function's output changes when you change the input by a small amount.
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ln a: The natural logarithm of a. The natural logarithm is the logarithm to the base e, where e is Euler's number (approximately 2.71828).
The Formula
The derivative of an exponential function f(x) = ax is:
f'(x) = ax ln a
Examples
Here are a couple of examples to illustrate:
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Example 1: If f(x) = 2x, then f'(x) = 2x ln 2.
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Example 2: If f(x) = 10x, then f'(x) = 10x ln 10.
Special Case: The Natural Exponential Function
A particularly important exponential function is the natural exponential function, f(x) = ex, where e is Euler's number. Since ln e = 1, the derivative of ex is simply ex:
- If f(x) = ex, then f'(x) = ex ln e = ex (1) = ex.
How to derive the formula
As the reference states, the derivative of an exponential function can be derived using the first principle of differentiation and limits.
Importance
Understanding the derivative of exponential functions is crucial in many areas of mathematics, science, and engineering, including:
- Calculus: Used in optimization problems, related rates, and understanding the behavior of functions.
- Physics: Modeling radioactive decay and population growth.
- Finance: Calculating compound interest and modeling investment growth.
