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User:Michael Hardy
d
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{\displaystyle {\begin{aligned}{\frac {dy}{dx}}={}&{\frac {d}{dx}}{\Big (}(2x+1)^{3}(3x-1)^{5}{\Big )}=(2x+1)^{3}\cdot {\frac {d}{dx}}(3x-1)^{5}+(3x-1)^{5}{\frac {d}{dx}}(2x+1)^{3}\\[12pt]={}&(2x+1)^{3}\cdot 5(3x-1)^{4}\cdot 3+(3x-1)^{3}\cdot 3(2x+1)^{2}\cdot 2\\[12pt]={}&15(2x+1)^{3}(3x-1)^{4}+6(3x-1)^{3}(2x+1)^{2}\\[12pt]={}&(3x-1)^{3}(2x+1)^{2}{\Big (}15(2x+2)(3x-1)+6{\Big )}\\[12pt]={}&6(3x-1)^{3}(2x+1)^{2}{\Big (}15x^{2}+10x-4{\Big )}\\[12pt]={}&6(3x-1)^{3}(2x+1)^{2}\cdot 15\left(x-{\tfrac {5+{\sqrt {85}}}{15}}\right)\left(x-{\tfrac {5-{\sqrt {85}}}{15}}\right)\\[12pt]={}&90\cdot 3^{3}\cdot 2^{2}\cdot \left(x-{\frac {1}{3}}\right)^{3}\left(x+{\frac {1}{2}}\right)^{2}\left(x-{\tfrac {5+{\sqrt {85}}}{15}}\right)\left(x-{\tfrac {5-{\sqrt {85}}}{15}}\right)\end{aligned}}}
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{\displaystyle {\begin{aligned}{}\\[8pt]{\frac {dy}{dx}}={}&{\frac {d}{dx}}\,{\frac {\tan x}{1+\cos x}}={\frac {(1+\cos x){\frac {d}{dx}}\tan x-(\tan x){\frac {d}{dx}}(1+\cos x)}{(1+\cos x)^{2}}}\\[12pt]={}&{\frac {(1+\cos x)\sec ^{2}x+(\tan x)\sin x}{(1+\cos x)^{2}}}\\[12pt]{}\end{aligned}}}
Problem #3 starts here:
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{\displaystyle {\begin{aligned}&xe^{y}=y\sin x\\[12pt]&x{\frac {d}{dx}}e^{y}+e^{y}{\frac {d}{dx}}x=y\left({\frac {d}{dx}}\sin x\right)+(\sin x){\frac {dy}{dx}}\\[12pt]&xe^{y}{\frac {dy}{dx}}+e^{y}=y\cos x+(\sin x){\frac {dy}{dx}}\\[12pt]&xe^{y}{\frac {dy}{dx}}-(\sin x){\frac {dy}{dx}}=y\cos x-e^{y}\\[12pt]&(xe^{y}-\sin x){\frac {dy}{dx}}=y\cos x-e^{y}\\[12pt]&{\frac {dy}{dx}}={\frac {y\cos x-e^{y}}{xe^{y}-\sin x}}\\[12pt]{}\end{aligned}}}
Problem #4 starts here:
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{\displaystyle {\begin{aligned}y={}&(\arcsin(2x))^{2}+\cot(\csc x)\\[12pt]{\frac {dy}{dx}}={}&2(\arcsin(2x))\cdot {\frac {d}{dx}}\arcsin(2x)-\csc ^{2}(\csc x)\cdot (-\csc x\cot x)\\[12pt]={}&2(\arcsin(2x))\cdot {\frac {1}{\sqrt {1-(2x)^{2}}}}\cdot {\frac {d}{dx}}(2x)+\csc ^{2}(\csc x)(\csc x\cot x)\\[12pt]={}&2(\arcsin(2x))\cdot {\frac {1}{\sqrt {1-(2x)^{2}}}}\cdot 2+\csc ^{2}(\csc x)(\csc x\cot x)\\[12pt]={}&2(\arcsin(2x))\cdot {\frac {1}{\sqrt {1-(2x)^{2}}}}\cdot 2+{\big (}\csc(\csc x){\big )}^{2}(\csc x\cot x)\\[12pt]={}&{\frac {4\arcsin(2x)}{\sqrt {1-(2x)^{2}}}}+{\big (}\csc(\csc x){\big )}^{2}(\csc x\cot x)\\[12pt]{}\end{aligned}}}
Problem #5 starts here:
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Recall that
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{\displaystyle {\begin{aligned}y={}&\tan(10^{x})+10^{\tan(\pi x)}\\[12pt]{\text{Recall that }}&{\frac {d}{dx}}10^{x}=10^{x}\cdot \log _{e}10\\[12pt]{\frac {dy}{dx}}={}&\sec ^{2}(10^{x})\cdot {\frac {d}{dx}}10^{x}+10^{\tan(\pi x)}\cdot {\frac {d}{dx}}\tan(\pi x)\\[12pt]={}&\sec ^{2}(10^{x})\cdot 10^{x}\cdot \ln 10+10^{\tan(\pi x)}\cdot \sec ^{2}(\pi x)\cdot {\frac {d}{dx}}(\pi x)\\[12pt]={}&\sec ^{2}(10^{x})\cdot 10^{x}\cdot \ln 10+10^{\tan(\pi x)}\cdot \sec ^{2}(\pi x)\cdot \pi \end{aligned}}}
![{\displaystyle {\begin{aligned}{\frac {dy}{dx}}={}&{\frac {d}{dx}}{\Big (}(2x+1)^{3}(3x-1)^{5}{\Big )}=(2x+1)^{3}\cdot {\frac {d}{dx}}(3x-1)^{5}+(3x-1)^{5}{\frac {d}{dx}}(2x+1)^{3}\\[12pt]={}&(2x+1)^{3}\cdot 5(3x-1)^{4}\cdot 3+(3x-1)^{3}\cdot 3(2x+1)^{2}\cdot 2\\[12pt]={}&15(2x+1)^{3}(3x-1)^{4}+6(3x-1)^{3}(2x+1)^{2}\\[12pt]={}&(3x-1)^{3}(2x+1)^{2}{\Big (}15(2x+2)(3x-1)+6{\Big )}\\[12pt]={}&6(3x-1)^{3}(2x+1)^{2}{\Big (}15x^{2}+10x-4{\Big )}\\[12pt]={}&6(3x-1)^{3}(2x+1)^{2}\cdot 15\left(x-{\tfrac {5+{\sqrt {85}}}{15}}\right)\left(x-{\tfrac {5-{\sqrt {85}}}{15}}\right)\\[12pt]={}&90\cdot 3^{3}\cdot 2^{2}\cdot \left(x-{\frac {1}{3}}\right)^{3}\left(x+{\frac {1}{2}}\right)^{2}\left(x-{\tfrac {5+{\sqrt {85}}}{15}}\right)\left(x-{\tfrac {5-{\sqrt {85}}}{15}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dafaaacd08decd2ca74fe75c17f2a065b5cceb6)
![{\displaystyle {\begin{aligned}{}\\[8pt]{\frac {dy}{dx}}={}&{\frac {d}{dx}}\,{\frac {\tan x}{1+\cos x}}={\frac {(1+\cos x){\frac {d}{dx}}\tan x-(\tan x){\frac {d}{dx}}(1+\cos x)}{(1+\cos x)^{2}}}\\[12pt]={}&{\frac {(1+\cos x)\sec ^{2}x+(\tan x)\sin x}{(1+\cos x)^{2}}}\\[12pt]{}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00895d80c360ca89e5d401c9790c44c356c992fa)
![{\displaystyle {\begin{aligned}&xe^{y}=y\sin x\\[12pt]&x{\frac {d}{dx}}e^{y}+e^{y}{\frac {d}{dx}}x=y\left({\frac {d}{dx}}\sin x\right)+(\sin x){\frac {dy}{dx}}\\[12pt]&xe^{y}{\frac {dy}{dx}}+e^{y}=y\cos x+(\sin x){\frac {dy}{dx}}\\[12pt]&xe^{y}{\frac {dy}{dx}}-(\sin x){\frac {dy}{dx}}=y\cos x-e^{y}\\[12pt]&(xe^{y}-\sin x){\frac {dy}{dx}}=y\cos x-e^{y}\\[12pt]&{\frac {dy}{dx}}={\frac {y\cos x-e^{y}}{xe^{y}-\sin x}}\\[12pt]{}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/295f428f3492b62db43a4dbf9b1e309358d87ea7)
![{\displaystyle {\begin{aligned}y={}&(\arcsin(2x))^{2}+\cot(\csc x)\\[12pt]{\frac {dy}{dx}}={}&2(\arcsin(2x))\cdot {\frac {d}{dx}}\arcsin(2x)-\csc ^{2}(\csc x)\cdot (-\csc x\cot x)\\[12pt]={}&2(\arcsin(2x))\cdot {\frac {1}{\sqrt {1-(2x)^{2}}}}\cdot {\frac {d}{dx}}(2x)+\csc ^{2}(\csc x)(\csc x\cot x)\\[12pt]={}&2(\arcsin(2x))\cdot {\frac {1}{\sqrt {1-(2x)^{2}}}}\cdot 2+\csc ^{2}(\csc x)(\csc x\cot x)\\[12pt]={}&2(\arcsin(2x))\cdot {\frac {1}{\sqrt {1-(2x)^{2}}}}\cdot 2+{\big (}\csc(\csc x){\big )}^{2}(\csc x\cot x)\\[12pt]={}&{\frac {4\arcsin(2x)}{\sqrt {1-(2x)^{2}}}}+{\big (}\csc(\csc x){\big )}^{2}(\csc x\cot x)\\[12pt]{}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9014472b83a75e8435287157242fcec255edd9c7)
![{\displaystyle {\begin{aligned}y={}&\tan(10^{x})+10^{\tan(\pi x)}\\[12pt]{\text{Recall that }}&{\frac {d}{dx}}10^{x}=10^{x}\cdot \log _{e}10\\[12pt]{\frac {dy}{dx}}={}&\sec ^{2}(10^{x})\cdot {\frac {d}{dx}}10^{x}+10^{\tan(\pi x)}\cdot {\frac {d}{dx}}\tan(\pi x)\\[12pt]={}&\sec ^{2}(10^{x})\cdot 10^{x}\cdot \ln 10+10^{\tan(\pi x)}\cdot \sec ^{2}(\pi x)\cdot {\frac {d}{dx}}(\pi x)\\[12pt]={}&\sec ^{2}(10^{x})\cdot 10^{x}\cdot \ln 10+10^{\tan(\pi x)}\cdot \sec ^{2}(\pi x)\cdot \pi \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a474697bd4d0499fa3d079502241c8c052e2432c)