$$C^\prime=0, C = const$$

$$(x^n)^\prime=nx^{n-1}$$

$$(a^x)^\prime=a^x\cdot\ln{a}$$

$$(e^x)^\prime=e^x$$

$$(\log_a{x})^\prime=\frac{1}{x\ln{a}}$$

$$(\ln{x})^\prime=\frac{1}{x}$$

$$(\sin{x})^\prime=\cos{x}$$

$$(\cos{x})^\prime=-\sin{x}$$

$$(\tg{x})^\prime=\frac{1}{\cos^2{x}}$$

$$(\ctg{x})^\prime=-\frac{1}{\sin^2{x}}$$

$$(\arcsin{x})^\prime=\frac{1}{\sqrt{1-x^2}}$$

$$(\arccos{x})^\prime=-\frac{1}{\sqrt{1-x^2}}$$

$$(\arctg{x})^\prime=\frac{1}{1+x^2}$$

$$(\arcctg{x})^\prime=-\frac{1}{1+x^2}$$