Reference Materials for Mathematics

\begin{align*} \lim_{x \rightarrow 0} \frac{\sin x}{x} \amp = \lim_{x \rightarrow 0} \frac{x}{\sin x} = 1 \\ \lim_{x \rightarrow 0^+} (1 + x)^{\frac{1}{x}} \amp = e \end{align*}

\begin{align*} f^\prime(a) \amp = \frac{df}{dx}(a) = \frac{df}{dx} \Big|_{x = a} = \lim_{x\rightarrow a} \frac{f(x) - f(a)}{x-a} = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \end{align*}

\begin{align*} \frac{d}{dx} x^r \amp = rx^{r-1} \amp \amp r \neq -1 \end{align*}

\begin{align*} \frac{d}{dx} (f+g) \amp = \frac{df}{dx} + \frac{dg}{dx} \\ \frac{d}{dx} (f-g) \amp = \frac{df}{dx} - \frac{dg}{dx}\\[1em] \\ \frac{d}{dx} cf \amp = c\frac{df}{dx} \end{align*}

\begin{align*} \frac{d}{dx} (fg) \amp = f \frac{dg}{dx} + g \frac{df}{dx} \end{align*}

\begin{align*} \frac{d}{dx} \frac{f}{g} \amp = \frac{ g \frac{df}{dx} - f \frac{dg}{dx}}{g^2} \end{align*}

\begin{align*} \frac{d}{dx} f(g(x)) \amp = f^\prime(g(x)) g^\prime(x) = \left. \frac{df(u)}{du} \right|_{u = g(x)} \frac{dg}{dx} \end{align*}

\begin{align*} \frac{d}{dx} f^{-1}(x) \amp = \frac{1}{f^\prime(f^{-1}(x))} \end{align*}

\begin{align*} \frac{d}{dx} e^x \amp = e^x \\ \frac{d}{dx} a^x \amp = a^x \ln a \\ \frac{d}{dx} \ln x \amp = \frac{1}{x} \\ \frac{d}{dx} \log_a x \amp = \frac{1}{x \ln a} \\ \frac{d}{dx} \sin x \amp = \cos x \\ \frac{d}{dx} \cos x \amp = - \sin x \\ \frac{d}{dx} \tan x \amp = \sec^2 x \\ \frac{d}{dx} \sec x \amp = \sec x \tan x \\ \frac{d}{dx} \csc x \amp = -\csc x \cot x \\ \frac{d}{dx} \cot x \amp = -\csc^2 x \\ \frac{d}{dx} \sinh x \amp = \cosh x \\ \frac{d}{dx} \cosh x \amp = \sinh x \\ \frac{d}{dx} \tanh x \amp = \sech^2 x \\ \frac{d}{dx} \sech x \amp = -\sech x \tanh x \\ \frac{d}{dx} \csch x \amp = -\csch x \coth x \\ \frac{d}{dx} \coth x \amp = -\csch^2 x \\ \frac{d}{dx} \arcsin x \amp = \frac{1}{\sqrt{1-x^2}} \\ \frac{d}{dx} \arccos x \amp = \frac{-1}{\sqrt{1-x^2}} \\ \frac{d}{dx} \arctan x \amp = \frac{1}{1+x^2} \\ \frac{d}{dx} \arcsec x \amp = \frac{1}{x\sqrt{x^2-1}} \\ \frac{d}{dx} \arccsc x \amp = \frac{-1}{x\sqrt{x^2-1}} \\ \frac{d}{dx} \arccot x \amp = \frac{-1}{1+x^2} \\ \frac{d}{dx} \arcsinh x \amp = \frac{1}{\sqrt{x^2+1}} \\ \frac{d}{dx} \arccosh x \amp = \frac{1}{\sqrt{x^2-1}} \\ \frac{d}{dx} \arctanh x \amp = \frac{1}{1-x^2} \\ \frac{d}{dx} \arcsech x \amp = \frac{-1}{x\sqrt{1-x^2}} \\ \frac{d}{dx} \arccsch x \amp = \frac{-1}{|x|\sqrt{x^2+1}} \\ \frac{d}{dx} \arccoth x \amp = \frac{1}{1-x^2} \end{align*}