Derivative

Exponential and logarithmic functions:

(1)
\begin{align} \frac{d}{dx}e^x = e^x \end{align}
(2)
\begin{align} \frac{d}{dx}a^x = \ln(a)a^x \end{align}
(3)
\begin{align} \frac{d}{dx}\ln(x) = \frac{1}{x},\qquad x > 0 \end{align}
(4)
\begin{align} \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)} \end{align}

Trigonometric function:

(5)
\begin{align} \frac{d}{dx}\sin(x) = \cos(x). \end{align}
(6)
\begin{align} \frac{d}{dx}\cos(x)= -\sin(x). \end{align}
(7)
\begin{align} \frac{d}{dx}\tan(x)= \sec^2(x). \end{align}

Inverse trigonometric function]]

(8)
\begin{align} \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}. \end{align}
(9)
\begin{align} \frac{d}{dx}\arccos(x)= -\frac{1}{\sqrt{1-x^2}}. \end{align}
(10)
\begin{align} \frac{d}{dx}\arctan(x)= \frac{1}{{1+x^2}}. \end{align}
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