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How is the derivative of a^x related to ln a?

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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, the exponential function is defined as $y = a^{x}$ , where $a$ is a positive real number not equal to 1. The derivative of this function with respect to $x$ is given by the formula:

$\frac{d}{d x} a^{x} = a^{x} \ln a$

In this relationship, $\ln a$ acts as a constant scaling factor. This result arises because any exponential function can be expressed in terms of the natural base $e$ as $a^{x} = e^{x \ln a}$ . Applying the chain rule to this expression confirms that the derivative of $a^{x}$ is the original function multiplied by the natural logarithm of the base, $\ln a$ .1

This contrasts with the natural exponential function $e^{x}$ , where the derivative is simply $e^{x}$ because $\ln e = 1$ .1