Derivative of Exponential
The exponential function is its own derivative.
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Example
Unique property of e
How it works
The function e^x is special because its rate of change at every point matches its current value, so differentiating it leaves it unchanged. This comes from the limit definition of the derivative and the fact that e is the base that makes the growth factor and instantaneous rate line up exactly. In practice, it is the simplest exponential function to differentiate because no extra constant appears.
When to use it
Use this rule whenever you need the slope or rate of change of an exponential model with base e, such as continuous growth, decay, compounding, or natural-log-based models. It is also a building block for differentiating more complicated expressions like e^(3x) or e^(x^2) using the chain rule.
Worked example, step by step
Differentiate f(x) = e^x at x = 2. Step 1: Use the derivative rule: f'(x) = e^x. Step 2: Substitute x = 2: f'(2) = e^2. Step 3: Approximate if needed: e^2 ≈ 7.389. Final answer: the slope at x = 2 is e^2 ≈ 7.389.
Common mistakes to avoid
- thinking the derivative of e^x is 0 because e is a constant
- forgetting that this rule is only for e^x, not for e^(ax) without the chain rule
- mixing up e^x with x^e and trying to apply the same derivative rule
Did You Know?
The function e^x is the only one that is its own derivative, which is what makes e so special in mathematics.
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