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