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First order derivatives

Find the derivatives of the following functions:

1. 

Solution.

 

Answer: 

 

2.                   

Solution.

 

Answer: 

 

3.        

Solution.

 

Answer

 

 

4.                       

Solution.

 

Answer:  

 

5. 

Solution.

 

Answer: .

 

6.                  

Solution.

 

Answer: 

 

7.                

Solution.

 

Answer: 

 

8.   

Solution.

 

Answer

 

9.    

Solution.

 

Answer

 

10. 

Solution.

 

Answer: .

 

11.           

Solution.

 

Answer

 

12.        

Solution.

 

Answer

 

13.  

Solution.

 

Answer

 

14.  

Solution.

Answer

 

15. 

Solution.

 

Answer: 

 

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: $y'=3\frac{tg^2 x}{\cos^2 x}.$