Derivatives of Exponential Functions of the Form \(c\cdot e^x\)
Deriving the Natural Exponential Function
The natural exponential function \(e^x\), where \(e=2.71828…\), is a curious function with a derivative that is equally interesting.
Yep. The derivative of the natural exponential function \(e^x\) is itself! Up to constant multiples of this function, it is the only function that has this property! Using this property along with the fact that derivatives can be taken term-by-term, we now have a larger class of functions whose derivatives we can find using easy rules! Try some of the examples in the “Examples” section below!
Use the fact that \(\frac{d}{dx}e^x=e^x\)
Derivative of the Natural Exponential: \(\frac{d}{dx}e^x=e^x\)
\(f'(x)=e^x+2\)
Use the fact that \(\frac{d}{dx}e^x=e^x\)
Derivative of the Natural Exponential: \(\frac{d}{dx}e^x=e^x\)
\(f'(x)=3e^x-e^x\) or \(f'(x)=2e^x\)
Use the fact that \(\frac{d}{dx}e^x=e^x\)
Derivative of the Natural Exponential: \(\frac{d}{dx}e^x=e^x\)
\(f'(x)=3.14e^x-2x+5e^x\) or \(f'(x)=9.14e^x-2x\)
Use the fact that \(\frac{d}{dx}e^x=e^x\)
Derivative of the Natural Exponential: \(\frac{d}{dx}e^x=e^x\)
\(f'(x)=2x+2e^x-3x^2\)
Use the fact that \(\frac{d}{dx}e^x=e^x\)
Derivative of the Natural Exponential: \(\frac{d}{dx}e^x=e^x\)
\(f'(x)=e^x+2e^x+3e^x+2020e^x\) or \(f'(x)=2026e^x\)