2024/G2/12: Difference between revisions
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1) <math>{\frac{d}{dx}} [5 + \pi] = 0+0=0 </math> <br><br> | 1) <math>{\frac{d}{dx}} [5 + \pi] = 0+0=0 </math> <br><br> | ||
2) <math>{\frac{d}{dx}} [3x] = 3 \cdot {\frac{d}{dx}} [x] = 3(1)(x)^{1-1} = 3 \cdot x^0 = 3 </math> <br><br> | 2) <math>{\frac{d}{dx}} [3x] = 3 \cdot {\frac{d}{dx}} [x] = 3(1)(x)^{1-1} = 3 \cdot x^0 = 3 </math> <br><br> | ||
3) <math>{\frac{d}{dx}} [9x^2] = 18x </math> <br><br> | 3) <math>{\frac{d}{dx}} [9x^2] = 18x </math> <br><br> | ||
<math> \sqrt(x) </math> | 4) <math>{\frac{d}{dx}} [x^3 + x^2 + 10] = 3x^2 +2x + 0 </math> <br><br> | ||
5) <math>{\frac{d}{dx}} [\sqrt(x)] = {\frac{d}{dx}} [x^{\frac{1}{2}}] = {\frac{1}{2x}}^{{\frac{1}{2}}-{\frac{2}{2}}} = {\frac{1}{2x}}^{\frac{-1}{2}} = {\frac{1}{2\sqrt(x)}}</math><br><br> | |||
A1. <math> {\sqrt[n](x^m)} = ({\sqrt[n]x})^m = x^{\frac{m}{n}} </math><br><br> | |||
6) <math> {\frac{d}{dx}} [{\sqrt[3](x^2)}] = {\frac{d}{dx}} [x^{\frac{2}{3}}] = {\frac{2}{3}} \cdot x^{\frac{-1}{3}} = {\frac{2}{3} \cdot \sqrt[3](x)} </math><br><br> | |||
7) <math> {\frac{d}{dx}} [(\sqrt[5]x)^7] = {\frac{d}{dx}} [{\frac{7}{x^5}}] = {\frac{7}{5}} \cdot x^{{\frac{7}{5}} - 1} = {\frac{7}{5}} \cdot x^{{\frac{2}{5}}} = {\frac{7}{5}} \cdot {\sqrt [5]x^2} </math><br><br> | |||
8) <math> {\frac{d}{dx}} [3x^{10}+e^{x}-5^{x}] = 30x^{9} + e^{x} - ln(5) \cdot 5^{x} </math><br><br> | |||
9) <math> {\frac{d}{dx}} [{\frac{1}{x}}] = {\frac{d}{dx}} [x^{-1}] = -1 \cdot x^{-1-1} = -x^{-2} = - {\frac{1}{x^2}} </math><br><br><br> | |||
<math>\mathbf{\color{Purple}{PracticeProblems}}</math><br> | |||
1) <math> {\frac{d}{dx}} [\pi + 3x -5x^{3}] = 0+3-15x^{2} </math><br><br> | |||
2) <math> {\frac{d}{dx}} [e^{x} + 3^{x} - 5 \cdot 2^{x}] = e^{x} + ln(3) \cdot 3^{x} -5 \cdot ln(2) \cdot 2^{x} </math><br><br> | |||
3) <math> {\frac{d}{dx}} [x^{5} - 6x^{3} +2x +5] = 5x^{4} -18x^{2} + 2 + 0 </math><br><br> | |||
4) <math> {\frac{d}{dx}} [x\sqrt x] = {\frac{d}{dx}} [x^{1} \cdot x^{{\frac{1}{2}}}] = {\frac{d}{dx}} [x^{{\frac{3}{2}}}] = {\frac{3}{2}} \cdot x^{{\frac{3}{2}} -1} = {\frac{3}{2x}}^{{\frac{3}{2}}} - {\frac{2}{2}} = {\frac{3}{2x}}^{{\frac{1}{2}}} = {\frac{3}{2}} \cdot \sqrt x </math><br><br> | |||
5) <math> {\frac{d}{dx}} [{\frac{3}{x}} + \sqrt 2 - 5^{x}] = 3 \cdot {\frac{d}{dx}} [{\frac{1}{x}}] = 3 \cdot {\frac{d}{dx}} [x^{-1}] = (-1)(3) \cdot x^{-1-1} = -3 \cdot x^{-2} = {\frac{3}{x^{2}}} + 0 - ln(-5) \cdot -5^{x} = -{\frac{3}{x^{2}}} - ln(-5) \cdot -5^{x} </math> <br><br> | |||
\sqrt{x} < | 6) <math> {\frac{d}{dx}} [{\frac{4}{\sqrt[3]x^{2}}} - 3x^{9} + \sqrt[3]x \cdot {\frac{1}{x}}] = 4 \cdot x^{-{\frac{2}{3}}} = x^{-{\frac{2}{3}}-{\frac{3}{3}}} = x^{-{\frac{5}{3}}} </math><br><br> | ||
Latest revision as of 16:45, 14 April 2023
1. The derivative of a constant is 0 =
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![{\displaystyle {\frac {d}{dx}}[c]=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/834381b512bc1d1fc47c2d26ce38ae71daaa1053)
![{\displaystyle {\frac {d}{dx}}[x^{n}]=nx^{n-1}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d3ba3abeaf2660e0d9e7591b7051715c53770112)
![{\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e100ac3ba73695d9c36ad8ada81efba4c2a01129)
![{\displaystyle {\frac {d}{dx}}[f(x)+g(x)]={\frac {d}{dx}}[f(x)]+{\frac {d}{dx}}[g(x)]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fff2f7dfe33f2ad22d0a3a717c191cf78c3286d7)
![{\displaystyle {\frac {d}{dx}}[a^{x}]=ln(a)\cdot a^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3fb6fb97381b39ec5218ede903245c8670575d12)
![{\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/783a58bf19ce36ca7781d7d31d78161a55406bff)
1)
![{\displaystyle {\frac {d}{dx}}[5+\pi ]=0+0=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dfece50fcfde546d6d76017b25135a8ec8de2fa9)
2)
![{\displaystyle {\frac {d}{dx}}[3x]=3\cdot {\frac {d}{dx}}[x]=3(1)(x)^{1-1}=3\cdot x^{0}=3}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/12ce4dd69616ec29e87b275a573a6503c7cec018)
3)
![{\displaystyle {\frac {d}{dx}}[9x^{2}]=18x}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f88eae1bfa924d1c2798c91554ebf3c9141d1c69)
4)
![{\displaystyle {\frac {d}{dx}}[x^{3}+x^{2}+10]=3x^{2}+2x+0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e7df9a7716c47d3c5b968d08176e825f9c7280e6)
5)
![{\displaystyle {\frac {d}{dx}}[{\sqrt {(}}x)]={\frac {d}{dx}}[x^{\frac {1}{2}}]={\frac {1}{2x}}^{{\frac {1}{2}}-{\frac {2}{2}}}={\frac {1}{2x}}^{\frac {-1}{2}}={\frac {1}{2{\sqrt {(}}x)}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1c411e38ef9f89182f66917c37201e6130664cf0)
A1.
![{\displaystyle {{\sqrt[{n}]{(}}x^{m})}=({\sqrt[{n}]{x}})^{m}=x^{\frac {m}{n}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ce6d6bedaca676fa7829acd50074e0944ba9e703)
6)
![{\displaystyle {\frac {d}{dx}}[{{\sqrt[{3}]{(}}x^{2})}]={\frac {d}{dx}}[x^{\frac {2}{3}}]={\frac {2}{3}}\cdot x^{\frac {-1}{3}}={{\frac {2}{3}}\cdot {\sqrt[{3}]{(}}x)}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9796dc4deb3c9107b1f71ac88d1ba4690768f8c5)
7)
![{\displaystyle {\frac {d}{dx}}[({\sqrt[{5}]{x}})^{7}]={\frac {d}{dx}}[{\frac {7}{x^{5}}}]={\frac {7}{5}}\cdot x^{{\frac {7}{5}}-1}={\frac {7}{5}}\cdot x^{\frac {2}{5}}={\frac {7}{5}}\cdot {{\sqrt[{5}]{x}}^{2}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/34b4847ab8363f555bd457a635ad5e8fa7fce59c)
8)
![{\displaystyle {\frac {d}{dx}}[3x^{10}+e^{x}-5^{x}]=30x^{9}+e^{x}-ln(5)\cdot 5^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e83248892037ceff9cf4ab579b30339746369d31)
9)
![{\displaystyle {\frac {d}{dx}}[{\frac {1}{x}}]={\frac {d}{dx}}[x^{-1}]=-1\cdot x^{-1-1}=-x^{-2}=-{\frac {1}{x^{2}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/09cd778938688031bb67da7c7b1d0280af0b2fde)
1)
![{\displaystyle {\frac {d}{dx}}[\pi +3x-5x^{3}]=0+3-15x^{2}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/48828e46be3f989f946aa536b1cdb21fe2951f4f)
2)
![{\displaystyle {\frac {d}{dx}}[e^{x}+3^{x}-5\cdot 2^{x}]=e^{x}+ln(3)\cdot 3^{x}-5\cdot ln(2)\cdot 2^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/bbbeada44288ec40b0b8b05864440e2cb48a9ed2)
3)
![{\displaystyle {\frac {d}{dx}}[x^{5}-6x^{3}+2x+5]=5x^{4}-18x^{2}+2+0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b3b04b00ea75f30e2904096be8148706fc2806e9)
4)
![{\displaystyle {\frac {d}{dx}}[x{\sqrt {x}}]={\frac {d}{dx}}[x^{1}\cdot x^{\frac {1}{2}}]={\frac {d}{dx}}[x^{\frac {3}{2}}]={\frac {3}{2}}\cdot x^{{\frac {3}{2}}-1}={\frac {3}{2x}}^{\frac {3}{2}}-{\frac {2}{2}}={\frac {3}{2x}}^{\frac {1}{2}}={\frac {3}{2}}\cdot {\sqrt {x}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5ed190e4bfaa36f04bfd1bb014904aaf06e6e46d)
5)
![{\displaystyle {\frac {d}{dx}}[{\frac {3}{x}}+{\sqrt {2}}-5^{x}]=3\cdot {\frac {d}{dx}}[{\frac {1}{x}}]=3\cdot {\frac {d}{dx}}[x^{-1}]=(-1)(3)\cdot x^{-1-1}=-3\cdot x^{-2}={\frac {3}{x^{2}}}+0-ln(-5)\cdot -5^{x}=-{\frac {3}{x^{2}}}-ln(-5)\cdot -5^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a06541ed18dd8515f5225e1db931b6831d0f644c)
6)
![{\displaystyle {\frac {d}{dx}}[{\frac {4}{{\sqrt[{3}]{x}}^{2}}}-3x^{9}+{\sqrt[{3}]{x}}\cdot {\frac {1}{x}}]=4\cdot x^{-{\frac {2}{3}}}=x^{-{\frac {2}{3}}-{\frac {3}{3}}}=x^{-{\frac {5}{3}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3cb687ec379dd92e633f72b01525ad60ca4d18f1)