The function whose derivative is itself is the exponential function, f(x) = ex.
Here's a breakdown:
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The Exponential Function: The exponential function, denoted as ex (where 'e' is Euler's number, approximately 2.71828), has a unique property in calculus.
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The Derivative: The derivative of a function represents its instantaneous rate of change.
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The Key Property: The derivative of ex is ex. In mathematical notation:
d/dx (ex) = ex
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Why this is important: This characteristic makes the exponential function crucial in various fields like:
- Differential Equations: Problems involving growth and decay often have exponential solutions.
- Physics: Radioactive decay, population growth, and other phenomena can be modeled using exponential functions.
- Finance: Compound interest calculations rely on exponential growth.
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Beyond ex: While ex is the most common example, any constant multiple of ex (i.e., Cex, where C is a constant) also has the property that its derivative is itself multiplied by that constant. For example, the derivative of 2ex is 2ex. Therefore, more precisely, the solution to the differential equation f'(x) = f(x) is Cex.
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Verification: We can verify this using the rules of differentiation.
- If f(x) = ex, then f'(x) = ex
- If f(x) = Cex, then f'(x) = Cex
The exponential function (ex) and its constant multiples are the functions whose derivative is equal to the function itself.
