4.6 Exponential and Logarithmic Derivatives

Notes

• $\displaystyle \frac{d}{dx}[e^x] = e^x$.
• $\displaystyle \frac{d}{dx}[\ln x] = \frac{1}{x}$ for $x > 0$.
• $\displaystyle \frac{d}{dx}[a^x] = a^x \ln a$. Example: $\displaystyle \frac{d}{dx}[2^x] = 2^x \ln 2$.
• Chain rule versions: $\displaystyle \frac{d}{dx}[e^{u}] = e^{u} u'$ and $\displaystyle \frac{d}{dx}[\ln u] = \frac{u'}{u}$.
• Example: $\displaystyle \frac{d}{dx}[e^{3x}] = 3e^{3x}$.