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# First order derivatives

## Find the derivatives of the following functions:

1.

Solution.

2.

Solution.

3.

Solution.

4.

Solution.

5.

Solution.

6.

Solution.

7.

Solution.

8.

Solution.

9.

Solution.

10.

Solution.

11.

Solution.

12.

Solution.

13.

Solution.

14.

Solution.

15.

Solution.

16. $y=(1+\ln\sin x)^2.$

Solution.

$$y'=((1+\ln\sin x)^2)'=2(1+\ln\sin x)(1+\ln\sin x)'=2(1+\ln\sin x)\frac{1}{\sin x}(\sin x)'=$$ $$=2(1+\ln\sin x)\frac{1}{\sin x}\cos x=2(1+\ln\sin x)ctg x.$$

Answer: $y'=2(1+\ln\sin x)ctg x.$

17. $y=3x^2+\sqrt[3]{x}-\frac{1}{x}+e^x+8.$

Solution.

$$y'=(3x^2+\sqrt[3]{x}-\frac{1}{x}+e^x+8)'=6x+\frac{1}{3}x^{-\frac{2}{3}}-(-x^{-2})+e^x=$$ $$=6x+\frac{1}{3\sqrt[3]{x^2}}+\frac{1}{x^{2}}+e^x.$$

Answer: $y'=6x+\frac{1}{3\sqrt[3]{x^2}}+\frac{1}{x^{2}}+e^x.$

18. $y=tg^3 x.$

Solution.

$$y'=(tg^3 x)'=3 tg^2 x(tg x)'=3tg^2 x\frac{1}{\cos^2 x}=3\frac{tg^2 x}{\cos^2 x}.$$

Answer: $y'=3\frac{tg^2 x}{\cos^2 x}.$