Have you ever wondered how to take the derivative of an exponential function like \(2^x\) or \(4^{3x^2}\)? In this article, we will derive the formula for taking the derivative of any exponential function whatsoever.
Derivation of the Formula
In order to derive the formula for the derivative of an exponential function, we first need to define a number \(e\) as the constant such that \(\frac{d}{dx}e^x = e^x\). Then we can find the derivative of any function \(a^x\), where \(a\) is a constant. Because \(a = e^{\ln a}\), we find that
\[ e^{\ln a^x} = e^{x\ln a} \]
and by the chain rule,
\[ \frac{dy}{dx} e^{x\ln a} = e^{x\ln a} \cdot \ln a = a^x \cdot \ln a \]
If we have some function of x as the exponent, we simply use the chain rule again:
\[ \frac{d}{dx} a^{f(x)} = a^{f(x)} \cdot \ln a \cdot f'(x) \]
And that’s it! Remember, if you ever get stuck on any problem involving exponential or logarithmic functions, try using log rules to manipulate the functions. It can be good to memorize the formulae, but it’s a lot better to understand the derivation of the formulae in case you forget them.
