\begin{aligned} 1.& \frac{dc}{dx}=0\\ &{c^\prime = 0}\\\\ 2.& \frac{d}{dx}(x^n) =  nx^{n-1} \\ & (x^n)^\prime = nx^{n-1}\\\\ 3.& \frac{d}{dx}(f+g) = \frac{df}{dx}+\frac{dg}{dx} \\ &(f+g)^\prime = f^\prime + g^\prime\\\\ 4.& \frac{d}{dx}(cf+kg) = c\frac{df}{dx}+k\frac{dg}{dx} \\ &(cf+kg)^\prime = cf^\prime + kg^\prime \\\\ 5.& \frac{d}{dx}(fg) = \frac{df}{dx}g+\frac{dg}{dx}f \\ &(fg)^\prime = f^{\prime}g + g^{\prime}f\\\\ 6.& \frac{d}{dx}(\frac{f}{g}) = \frac{g(x)\frac{d}{dx}f(x)-f(x)\frac{d}{dx}g(x)}{g(x)^2} \\ &(\frac{f}{g})^\prime = \frac{gf^{\prime} – fg^{\prime}}{g^2}\\\\ 7.& \frac{d}{dx}(u^n) = nu^{n-1}\frac{du}{dx}\\\\ 8.& \frac{d}{dx}(f \circ g) = \frac{d}{dg(x)}f(g(x))\frac{d}{dx}g(x) \\ & (f \circ g)^\prime (x) = f^{\prime}(g(x))g^\prime(x)\\\\ 9.& \frac{d}{dx}\ln|x| = \frac{1}{x} \\\\ 10.& \frac{d}{dx}e^x = e^x \\\\ 11.& \frac{d}{dx}a^x = a^x \in a \\\\ 12.& \frac{d}{dx}\log_a|x| = \frac{1}{x \in a} \\\\ 13.& \frac{d}{dx}\sin x = \cos x \\\\ 14.& \frac{d}{dx} \cos x = -\sin x \\\\ 15.& \frac{d}{dx}\tan x = \sec^2 x \\\\ 16.& \frac{d}{dx} \cot x = -\cosec^2 x \\\\ 17.& \frac{d}{dx}\sec x = \sec x \tan x \\\\ 18.& \frac{d}{dx} \cosec x = -\cosec x \cot x \\\\ 19.& \frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}} \\\\ 20.& \frac{d}{dx} \arccos x = – \frac{1}{\sqrt{1-x^2}} \\\\ 21.& \frac{d}{dx}\arctan x = \frac{1}{1+x^2} \\\\ 22.& \frac{d}{dx} \text{arccot} x = -\frac{1}{1-x^2} \\\\ 23.& \frac{d}{dx}\text{arcsec} x = \frac{1}{|x|\sqrt{x^2-1}}\\\\ 24.& \frac{d}{dx} \text{arccosec} x = – \frac{1}{|x|\sqrt{x^2-1}}\\\\ \end{aligned}