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How does the derivative of e^x compare with e^x itself?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

In mathematics, a key property of the natural exponential function, defined as $f (x) = e^{x}$ , is that its derivative is identical to the function itself.12

This relationship is expressed as:
$\frac{d}{d x} e^{x} = e^{x}$

Because the derivative represents the rate of change or the slope of the function at any given point, this equality indicates that the slope of the curve $e^{x}$ at any value of $x$ is exactly equal to the value of the function at that same point. This unique property distinguishes the natural exponential function from other exponential functions of the form $a^{x}$ , where the derivative is given by $\frac{d}{d x} a^{x} = a^{x} \ln a$ .23