umatisi6ar

2023-03-11

What is the derivative of $f\left(x\right)={b}^{x}$?

Edward Hunter

This is the exponential function of base b (where $b>0$ should be assumed). It can be thought of as ${b}^{x}={e}^{x\mathrm{ln}\left(b\right)}$, therefore, using the Chain Rule (See Chain Rule ) and the fact that $\left({e}^{x}\right)\prime ={e}^{x}$ (see Exponentials with Base e ) yields $\left({b}^{x}\right)\prime ={e}^{x\mathrm{ln}\left(b\right)}×\mathrm{ln}\left(b\right)={b}^{x}×\mathrm{ln}\left(b\right)$