Exponential and Logarithmic Derivatives

$\frac{d}{dx}[e^x] = e^x$, $\frac{d}{dx}[e^{u}] = e^u \cdot u'$

$\frac{d}{dx}[\ln x] = \frac{1}{x}$, $\frac{d}{dx}[\ln u] = \frac{u'}{u}$

$\frac{d}{dx}[a^x] = a^x \ln a$, $\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$

For $y = x^x$: take $\ln$ of both sides.

$\ln y = x \ln x$ → $\frac{y'}{y} = \ln x + 1$ → $y' = x^x(\ln x + 1)$.

$y = e^{3x^2}$ → $y' = 6xe^{3x^2}$.

$y = \ln(\sin x)$ → $y' = \frac{\cos x}{\sin x} = \cot x$.