|
|
Line 88: |
Line 88: |
| * [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Derivative%20Formulas/Presentation5b_Exponential%20and%20Logs.pptx Exponential and Logarithms]. PowerPoint file created by Professor Cynthia Roberts, UTSA. | | * [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Derivative%20Formulas/Presentation5b_Exponential%20and%20Logs.pptx Exponential and Logarithms]. PowerPoint file created by Professor Cynthia Roberts, UTSA. |
| | | |
− | ==Licensing=== | + | ==Licensing== |
| Content obtained and/or adapted from: | | Content obtained and/or adapted from: |
| * [https://en.wikibooks.org/wiki/Calculus/Tables_of_Derivatives Table of Derivatives, Wikibooks: Calculus] under a CC BY-SA license | | * [https://en.wikibooks.org/wiki/Calculus/Tables_of_Derivatives Table of Derivatives, Wikibooks: Calculus] under a CC BY-SA license |
General Rules
Powers and Polynomials
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(c)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ce780c607ce448b2c998b428102516d54ba069)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb301ebf75d19841c50d51254005cba1c8e1b33)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}x^{n}=nx^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a1da66e8edee4d6d452933563c5efa5289bb76)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae7062c9ba24f08ce669de60848d2476ab3bcdd)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16d6ec69f1e80d95455b75eacad19a1c0d6b33b0)
![{\displaystyle {{\frac {\mathrm {d} }{\mathrm {d} x}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e1e6eb210ba7c8cfa6fab2cbe7474e0b381bd6)
Trigonometric Functions
Exponential and Logarithmic Functions
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b91476b5bc58a251e75b2ff2c9a707bd03705)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}a^{x}=a^{x}\ln(a)\qquad {\text{if }}a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d00bbe193f09616a8663d54f924bd08269b5b3)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e0f2d4a2ba2eaa4aaea0752486212d65af14c3)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\log _{a}(x)={\frac {1}{x\ln(a)}}\qquad {\text{if }}a>0\ ,\ a\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8e596d3260f4d3fa3d1ec57200ef9d665a5990)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(f^{g})={\frac {\mathrm {d} }{\mathrm {d} x}}\left(e^{g\ln(f)}\right)=f^{g}\left(f'{\frac {g}{f}}+g'\ln(f)\right)\ ,\qquad f>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e122a0540f15017b528e2f2afdcec4c60f4323bb)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(c^{f})={\frac {\mathrm {d} }{\mathrm {d} x}}\left(e^{f\ln(c)}\right)=c^{f}\ln(c)\cdot f'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e45f30e76c06eba38130ca4700082400564b6de5)
Inverse Trigonometric Functions
Hyperbolic and Inverse Hyperbolic Functions
Resources
Licensing
Content obtained and/or adapted from: