\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}
\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}
