Table of Derivatives of Composite Functions.

$$c^{\prime }=0,c=const;$$

$$(u(x{)}^{\alpha }{)}^{\prime }=\alpha u^{\alpha -1}(x)u^{\prime }(x),x\in \mathbb{R},\alpha \in \mathbb{R};$$

$$(a^{u(x)}{)}^{\prime }=a^{u(x)}\ln a\cdot u^{\prime }(x),a>0,a\ne 1,x\in \mathbb{R};$$

$$(e^u(x){)}^{\prime }=e^{u(x)}{u}^{\prime }(x);$$

$$({\log }_{a}u(x){)}^{\prime }=\frac{1}{u(x)\ln a}{u}^{\prime }(x),x>0;$$

$$({\log }_a|u(x)|{)}^{\prime }=\frac{1}{u(x)\ln a}{u}^{\prime }(x),x\ne 0;$$

$$(\ln u(x){)}^{\prime }=\frac{1}{u(x)}{u}^{\prime }(x),x>0;$$

$$(\sin u(x){)}^{\prime }=\cos u(x)\cdot u^{\prime }(x),x\in \mathbb{R};$$

$$(\cos u(x){)}^{\prime }=-\sin u(x)\cdot u^{\prime }(x)x\in \mathbb{R};$$

$$(\mathrm{tg}{u}^{\prime }(x){)}^{\prime }=\frac{1}{{\cos }^2{u}^{\prime }(x)}{u}^{\prime }(x),x\ne \frac{\pi }{2}(2n+1),\in \mathbb{Z};$$

$$(\mathrm{ctg}u(x){)}^{\prime }=-\frac{1}{{\sin }^{2}u(x)}{u}^{\prime }(x),x\ne \pi n,n\in \mathbb{Z};$$

$$(\arcsin u(x){)}^{\prime }=\frac{1}{\sqrt{1-u^2(x)}}{u}^{\prime }(x),|x|<1;$$

$$(\arccos u(x){)}^{\prime }=-\frac{1}{\sqrt{1-u^2(x)}}{u}^{\prime }(x),|x|<1;$$

$$(\mathrm{arctg}u(x){)}^{\prime }=\frac{1}{1+u^2(x)}{u}^{\prime }(x),x\in \mathbb{R};$$

$$(\mathrm{arcctg}u(x){)}^{\prime }=-\frac{1}{1+u^2(x)}{u}^{\prime }(x),x\in \mathbb{R};$$

$$(\mathrm{sh}u(x){)}^{\prime }=\mathrm{ch}u(x)\cdot u^{\prime }(x),x\in \mathbb{R};$$

$$(\mathrm{ch}u(x){)}^{\prime }=\mathrm{sh}u(x)\cdot u^{\prime }(x)x\in \mathbb{R};$$

$$(\mathrm{th}u(x){)}^{\prime }=\frac{1}{{\mathrm{ch}}^{2}u(x)}{u}^{\prime }(x),x\ne \frac{\pi }{2}(2n+1),n\in \mathbb{Z};$$

$$(\mathrm{cth}u(x){)}^{\prime }=-\frac{1}{{\mathrm{sh}}^{2}u(x)}{u}^{\prime }(x),x\ne \pi n,n\in \mathbb{Z};$$

$$(\mathrm{arsh}u(x){)}^{\prime }=\frac{1}{\sqrt{u^2(x)+1}}{u}^{\prime }(x),x\in \mathbb{R};$$

$$(\mathrm{arch}u(x){)}^{\prime }=-\frac{1}{\sqrt{u^2(x)-1}}{u}^{\prime }(x),|x|>1;$$

$$(\mathrm{arth}u(x){)}^{\prime }=\frac{1}{1-u^2(x)}{u}^{\prime }(x),x\in \mathbb{R};$$

$$(\mathrm{arcth}u(x){)}^{\prime }=\frac{1}{u^2(x)-1}{u}^{\prime }(x),x\in \mathbb{R}.$$

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