Calculate the derivative of the exponential function base 10.

The derivative of the exponential function base 10 is equal to the natural logarithm of 10 multiplied by the function itself.

The exponential function base 10 is defined as f(x) = 10^x. To find its derivative, we use the formula for the derivative of an exponential function:

f'(x) = ln(a) * a^x

where ln(a) is the natural logarithm of the base a. In this case, a = 10, so we have:

f'(x) = ln(10) * 10^x

Therefore, the derivative of the exponential function base 10 is equal to ln(10) multiplied by the function itself. This means that the rate of change of the function at any point x is proportional to the function itself, with a constant of proportionality equal to ln(10).

We can also express this result in terms of the common logarithm (base 10) by using the change of base formula:

ln(10) = log(10) / ln(e)

where ln(e) is equal to 1. Therefore, we have:

f'(x) = log(10) * 10^x / ln(e)

Simplifying this expression, we get:

f'(x) = log(10) * 2.3026 * 10^x

So the derivative of the exponential function base 10 can also be expressed as log(10) multiplied by 2.3026 and the function itself.