$${\frac{d}{dx} \left(x^n \right) = nx^{n-1}}$$
$${\frac{d}{dx} \left(uv \right) = u'v + uv'}$$
$${\frac{d}{dx} \left(\frac{u}{v} \right) = \frac{u'v - uv'}{v^2}}$$
$${\frac{d}{dx} \left(f(g(x)) \right) = f'(g(x)) g'(x)}$$
$${\frac{d}{dx} (sin (x)) = cos(x)}$$
$${\frac{d}{dx} (cos (x)) = -sin(x)}$$
$${\frac{d}{dx} (tan (x)) = sec^2(x)}$$
$${\frac{d}{dx} (sec (x)) = sec(x)tan(x)}$$
$${\frac{d}{dx} (cot (x)) = -csc^2(x)}$$
$${\frac{d}{dx} (csc (x)) = -csc(x)cot(x)}$$
$${\frac{d}{dx} (sinh (x)) = cosh(x)}$$
$${\frac{d}{dx} (cosh (x)) = sinh(x)}$$
$${\frac{d}{dx} (tanh (x)) = sech^2(x)}$$
$${\frac{d}{dx} (coth (x)) = -csch^2(x)}$$
$${\frac{d}{dx} (sech (x)) = -sech(x) tanh(x)}$$
$${\frac{d}{dx} (csch (x)) = -csch(x) coth(x)}$$
$${\frac{d}{dx}(sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}}}$$
$${\frac{d}{dx}(tan^{-1}(x)) = \frac{1}{1 + x^2}}$$
$${\frac{d}{dx}(sec^{-1}(x)) = \frac{1}{x \sqrt{x^2 - 1}}}$$
$${\frac{d}{dx}(cos^{-1}(x)) = \frac{-1}{\sqrt{1 - x^2}}}$$
$${\frac{d}{dx}(cot^{-1}(x)) = \frac{-1}{1 + x^2}}$$
$${\frac{d}{dx}(csc^{-1}(x)) = \frac{-1}{x \sqrt{x^2 - 1}}}$$
$${\frac{d}{dx}(sinh^{-1}(x)) = \frac{1}{\sqrt{1 + x^2}}}$$
$${\frac{d}{dx}(cosh^{-1}(x)) = \frac{1}{\sqrt{x^2 - 1}}}$$
$${\frac{d}{dx}(tanh^{-1}(x)) = \frac{1}{1 - x^2}}$$