We shall first look at the irrational number $e$ in order to show its special properties when used with derivatives of exponential and logarithm functions. As mentioned before in the Algebra section, the value of $e$ is approximately $e\approx 2.718282$ but it may also be calculated as the Infinite Limit:
Here we could take the natural logarithm outside the limit because it doesn't have anything to do with the limit (we could have chosen not to). We then substituted the value of $e$ .
Derivative of the Natural Logarithm
${\frac {d}{dx}}\ln(x)={\frac {1}{x}}$
If we wanted, we could go through that same process again for a generalized base, but it is easier just to use properties of logs and realize that:
$\log _{a}(x)={\frac {\ln(x)}{\ln(a)}}$
Since ${\frac {1}{\ln(a)}}$ is a constant, we can just take it outside of the derivative:
Now that we have derived a specific case, let us extend things to the general case. Assuming that $a$ is a positive real constant, we wish to calculate:
${\frac {d}{dx}}a^{x}$
One of the oldest tricks in mathematics is to break a problem down into a form that we already know we can handle. Since we have already determined the derivative of $e^{x}$ , we will attempt to rewrite $a^{x}$ in that form.
Using that $e^{\ln(c)}=c$ and that $\ln(a^{b})=b\cdot \ln(a)$ , we find that:
We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such a products with many terms, quotients of composed functions, or functions with variable or function exponents. We do this by taking the natural logarithm of both sides, re-arranging terms using the logarithm laws below, and then differentiating both sides implicitly, before multiplying through by $y$ .