From Mr. V Wiki Math
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| 3) <math>{\frac{d}{dx}} [9x^2] = 18x </math> <br><br> | | 3) <math>{\frac{d}{dx}} [9x^2] = 18x </math> <br><br> |
| 4) <math>{\frac{d}{dx}} [x^3 + x^2 + 10] = 3x^2 +2x + 0 </math> <br> | | 4) <math>{\frac{d}{dx}} [x^3 + x^2 + 10] = 3x^2 +2x + 0 </math> <br> |
| 5) <math>{\frac{d}{dx}} [\sqrt(x)] = {\frac{d}{dx}} [x^{{\frac{1}{2}}] = {\frac{1}{2x}^{{\frac{1}{2}-1}} </math> <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> |
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| | <math> \sqrt(x) </math> |
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| | | \sqrt{x} <br><br><br><br><br><br><br><br><br> |
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| <math>{\frac{d}{dx}} [c] = 0 </math> <br> <math>{\frac{d}{dx}} [c\cdot f(x)] = c\cdot{\frac{d}{dx}} [f(x)] </math> <br> <math>{\frac{d}{dx}} [f(x)\pm g(x)] = {\frac{d}{dx}} [f(x)] \pm {\frac{d}{dx}} [g(x)] </math> <br> <math>{\frac{d}{dx}} [a^x] = \ln(a)a^x </math><br> <math>{\frac{d}{dx}} [e^x] = e^x </math><br> <math>\color{Blue}Power\,Rule </math><br> <math>{\frac{d}{dx}} [x^n] = n \cdot x^n-1 </math> <br> <math>\color{Red}Product\,Rule </math><br> <math>{\frac{d}{dx}} [f\cdot{g}]= {\frac{d}{dx}}[f]\cdot{g}+{\frac{d}{dx}}[g]\cdot{f}</math><br> <math>\color{Green}Quotient\,Rule </math><br> <math>{\frac{d}{dx}}[\frac{f}{g}]=\frac{{\frac{d}{dx}}[f]\cdot{g}-{\frac{d}{dx}}[g]\cdot{f}}{g^2}</math><br> <math>\mathbf{\color{Purple}{Examples}}</math><br> <math>\mathbf{Ex.1}</math><br> <math>if\,f(x)=x\cdot{e^x}</math><br> <math>f^\prime(x)=1\cdot{e^x}+x\cdot{e^x}</math><br> <math>\mathbf{Ex.2}</math><br> <math>if\,f(t)=\sqrt{t}(a+bt)</math><br> <math>f^\prime(t)=\frac{1}{2\sqrt{t}}(a+bt)+t\sqrt{t}(b)</math><br> <math>\mathbf{Ex.3}</math><br> <math>if\,f(x)=\sqrt{x}\cdot{g(x)}</math><br> <math>g(4)=2</math><br> <math>g^\prime(4)=3</math><br> <math>f^\prime(x)=\frac{1}{2\sqrt{x}}\cdot{g(x)}+\sqrt{x}\cdot{g^\prime(x)}</math><br> <math>\mathbf{Ex.4}</math><br> <math>y=\frac{\color{Blue}{x^2+x-2}}{\color{Red}{x^3+6}}</math><br> <math>{\frac{d}{dx}}=y^\prime=\frac{(2x+1)(x^3-6)-\color{Blue}{(x^2+x-2)}(3x^2)}{\color{Red}{(x^3+6)^2}}</math><br> <math>=\frac{(2x^4+x^4+x^3+12x+6-[3x^4+3x^2-6x^2]}{(x^3+6)^2}</math><br> <math>=\frac{-x^4-2x^3+6x^2+12x+6}{(x^3+6)^2}</math><br> <math>\mathbf{Ex.5}</math><br> <math>y=\frac{e^x}{1+x^2}\,(1,\frac{e}{2})\,</math><br> <math>{\frac{d}{dx}}=\frac{e^x\cdot(1+x^2)-e^x(2x)}{(1+x^2)^2}</math><br> <math>{\frac{d}{dx}}|_{x=1}\frac{e(1+1)-e^\prime(2)}{(1+1)^2}=\frac{2e-2e}{2^2}=\frac{0}{4}=0</math>
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Revision as of 17:38, 11 April 2023
![{\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)
![{\displaystyle {\frac {d}{dx}}[5+\pi ]=0+0=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dfece50fcfde546d6d76017b25135a8ec8de2fa9)
![{\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)
![{\displaystyle {\frac {d}{dx}}[9x^{2}]=18x}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f88eae1bfa924d1c2798c91554ebf3c9141d1c69)
![{\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)
![{\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)
