2024/G9/12: Difference between revisions
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<math>{\frac{d}{dx}} [a^x] = \ln(a)a^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>{\frac{d}{dx}} [e^x] = e^x </math><br> | ||
<math>\color{Blue}Power Rule </math><br> | <math>\color{Blue}Power\,Rule </math><br> | ||
<math>{\frac{d}{dx}} [x^n] = n \cdot x^n-1 </math> <br> | <math>{\frac{d}{dx}} [x^n] = n \cdot x^n-1 </math> <br> | ||
<math>\color{ | <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>{\frac{d}{dx}} [f\cdot{g}]= {\frac{d}{dx}}[f]\cdot{g}+{\frac{d}{dx}}[g]\cdot{f}</math><br> | ||
<math>\color{ | <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>{\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{Ex.1}</math><br> | <math>\mathbf{Ex.1}</math><br> | ||
<math>f(x)=x\cdot{e^x}</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>f^\prime(x)=1\cdot{e^x}+x\cdot{e^x}</math><br> | ||
<math>\mathbf{Ex.2}</math><br> | <math>\mathbf{Ex.2}</math><br> | ||
<math>f(t)=\sqrt{t}(a+bt)</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>f^\prime(t)=\frac{1}{2\sqrt{t}}(a+bt)+t\sqrt{t}(b)</math><br> | ||
<math>\mathbf{Ex.3}</math><br> | <math>\mathbf{Ex.3}</math><br> | ||
<math>if\,f(x)=\sqrt{x}\cdot{g(x)}</math><br> | <math>if\,f(x)=\sqrt{x}\cdot{g(x)}</math><br> | ||
<math>g(4)=2</math><br> | <math>g(4)=2</math><br> | ||
<math>g^\prime(4)=3</math> | <math>g^\prime(4)=3</math><br> | ||
<math>f^\prime(x)= | <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{x^2+x-2}{x^3+6}</math><br> | |||
<math>{\frac{d}{dx}}=y^\prime=\frac{(2x+1)(x^3-6)-(x^2+x-2)(3x^2)}{(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> | |||
Revision as of 18:08, 29 March 2023
![{\displaystyle {\frac {d}{dx}}[c]=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/834381b512bc1d1fc47c2d26ce38ae71daaa1053)
![{\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)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/27fc302c62ab11c846d1255170f40351d3461dfe)
![{\displaystyle {\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/63daf7259a950f7e09db6f16b13e760574c26bf7)
![{\displaystyle {\frac {d}{dx}}[e^{x}]=e^{x}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/783a58bf19ce36ca7781d7d31d78161a55406bff)
![{\displaystyle {\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/02df667a9de22b7a499cb1b37b3df322b3bfc5a4)
![{\displaystyle {\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/57ee97cfd2a83cef00e5454484f44a5c99a73190)
![{\displaystyle {\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/33117095dc4fc9bab8b77262a36eac752e1beaa5)
![{\displaystyle ={\frac {(2x^{4}+x^{4}+x^{3}+12x+6-[3x^{4}+3x^{2}-6x^{2}]}{(x^{3}+6)^{2}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fdb5247377f2d161c9bc989a350a4642c753c895)
