Derivatives of Exponential Functions. Class 12 Math.

The derivative of an exponential function \(f(x)=a^{x}\) is \(f^{\prime }(x)=a^{x}\ln (a)\), where \(\ln (a)\) is the natural logarithm of the base \(a\). A special case is the natural exponential function, \(f(x)=e^{x}\), whose derivative is itself, \(f^{\prime }(x)=e^{x}\), because the natural logarithm of \(e\) is 1. 
General rule
Formula:
The derivative of \(f(x)=a^{x}\) is \(f^{\prime }(x)=a^{x}\ln (a)\). 
How it works:
You keep the original function, \(a^{x}\), and multiply it by the natural logarithm of the base, \(\ln (a)\).
  Example:
The derivative of \(f(x)=5^{x}\) is \(f^{\prime }(x)=5^{x}\ln (5)\).  Natural exponential function
Formula:
The derivative of \(f(x)=e^{x}\) is \(f^{\prime }(x)=e^{x}\).  Why it's simple: This is a special case of the general rule because the natural logarithm of \(e\) (\(\ln (e)\)) is equal to 1, so the formula becomes \(f^{\prime }(x)=e^{x}\cdot 1\), which simplifies to \(f^{\prime }(x)=e^{x}\).