A.3: Таблиця похідних
- Page ID
- 60938
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По всій цій таблиці\(a\) і\(b\) знаходяться константи, незалежні від\(x\text{.}\)
| \(F(x)\) | \(F'(x)=\dfrac{dF}{dx}\) |
| \(af(x)+bg(x)\) | \(af'(x)+bg'(x)\) |
| \(f(x)+g(x)\) | \(f'(x)+g'(x)\) |
| \(f(x)-g(x)\) | \(f'(x)-g'(x)\) |
| \(af(x)\) | \(af'(x)\) |
| \(f(x)g(x)\) | \(f'(x)g(x)+f(x)g'(x)\) |
| \(f(x)g(x)h(x)\) | \(f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)\) |
| \(\frac{f(x)}{g(x)}\) | \(\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\) |
| \(\frac{1}{g(x)}\) | \(-\frac{g'(x)}{g(x)^2}\) |
| \(f\big(g(x)\big)\) | \(f'\big(g(x)\big)g'(x)\) |
| \(F(x)\) | \(F'(x)=\dfrac{dF}{dx}\) |
| \(a\) | \(0\) |
| \(x^a\) | \(ax^{a-1}\) |
| \(g(x)^a\) | \(ag(x)^{a-1}g'(x)\) |
| \(\sin x\) | \(\cos x\) |
| \(\sin g(x)\) | \(g'(x)\cos g(x)\) |
| \(\cos x\) | \(-\sin x\) |
| \(\cos g(x)\) | \(-g'(x)\sin g(x)\) |
| \(\tan x\) | \(\sec^2 x\) |
| \(\csc x\) | \(-\csc x\cot x\) |
| \(\sec x\) | \(\sec x\tan x\) |
| \(\cot x\) | \(-\csc^2 x\) |
| \(e^x\) | \(e^x\) |
| \(e^{g(x)}\) | \(g'(x)e^{g(x)}\) |
| \(a^x\) | \((\ln a)\ a^x\) |
| \(F(x)\) | \(F'(x)=\dfrac{dF}{dx}\) |
| \(\ln x\) | \(\frac{1}{x}\) |
| \(\ln g(x)\) | \(\frac{g'(x)}{g(x)}\) |
| \(\log_a x\) | \(\frac{1}{x\ln a}\) |
| \(\arcsin x\) | \(\frac{1}{\sqrt{1-x^2}}\) |
| \(\arcsin g(x)\) | \(\frac{g'(x)}{\sqrt{1-g(x)^2}}\) |
| \(\arccos x\) | \(-\frac{1}{\sqrt{1-x^2}}\) |
| \(\arctan x\) | \(\frac{1}{1+x^2}\) |
| \(\arctan g(x)\) | \(\frac{g'(x)}{1+g(x)^2}\) |
| \(\textrm{arccsc}\, x\) | \(-\frac{1}{|x|\sqrt{x^2-1}}\) |
| \(\textrm{arcsec}\, x\) | \(\frac{1}{|x|\sqrt{x^2-1}}\) |
| \(\textrm{arccot}\,x\) | \(-\frac{1}{1+x^2}\) |