Derivative of e^x
The exponential function is its own derivative.
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Example
e^x is its own derivative — at every point its slope equals its height.
How it works
The function e^x has a special property: when you measure its instantaneous rate of change, you get the same function back. This comes from the limit definition of the derivative, where the change in e^x over a tiny step h simplifies to e^x times the limit of (e^h - 1)/h, and that limit equals 1. So the slope of the graph at any x-value is exactly e^x itself.
When to use it
Use this rule whenever you need the slope of an exponential growth or decay expression written with base e. It is especially useful in calculus, physics, biology, and economics when modeling continuously changing quantities.
Worked example, step by step
Find the derivative of f(x) = e^x at x = 2. Step 1: Differentiate the function. f'(x) = d/dx(e^x) = e^x Step 2: Substitute x = 2. f'(2) = e^2 Step 3: Evaluate the value. e^2 \approx 7.389 Final answer: The slope at x = 2 is e^2 \approx 7.389.
Common mistakes to avoid
- Thinking the derivative of e^x is x e^(x-1) like a power rule expression
- Forgetting that the derivative of e^x is exactly e^x, with no extra coefficient
- Using this rule for e^(ax) without applying the chain rule to get a factor of a
Did You Know?
e^x is the only function that is its own derivative — the property that makes e so special.
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