高等数学-求导公式与法则
求导公式与法则
求导基础公式
[(x^a)'= ax^{a-1}
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(sqrt{x})'=frac{1}{2sqrt{x}}
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(frac{1}{x})'=-frac{1}{x^2}
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(a^x)'=a^xln{a}
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(log_a{x})'=frac{1}{xln{a}}
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(sin{x})'=cos{x}
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(cos{x})'=-sin{x}
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( an{x})'=sec^2{x}
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(cot{x})'=-csc^2{x}
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(sec{x})'=sec{x} an{x}
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(csc{x})'=-csc{x}cot{x}
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(arcsin{x})'=frac{1}{sqrt{1-x^2}}
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(arccos{x})'=-frac{1}{sqrt{1-x^2}}
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(arctan{x})'=frac{1}{1+x^2}
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(arccot{x})'=-frac{1}{1+x^2}
]
求导运算法则
设$ u(x)、v(x)$可导，则
| 四则求导法则 | 四则求微分法则 |
|---|---|
| $$ (upm v)'=u'pm v'$$ | $$d(upm v) = dupm dv$$ |
| $$ (1)(uv)'=u'v+v'u (2)(ku)'=ku'(k为常数) (3)(uvw)'=u'vw+uv'w+uvw'$$ | $$(1)d(uv)=udv+vdu (2)d(ku)=kdu(k为常数) (3)d(uvw)=vwdu+uwdv+uvdw$$ |
| $$(frac{u}{v})'=frac{u'v-uv'}{v^2}$$ | $$d(frac{u}{v})=frac{vdu-udv}{v^2}$$ |
复合函数求导法则-链式法则
设(y=f(u))可导，(u=phi(x))可导，且(phi^{'}(x) eq0)，则(y=f[phi(x)])可导，且$$frac{dy}{dx}=frac{dy}{du}.frac{du}{dx} = f{'}(u).phi{'}(x)= f{'}[phi(x)].phi{'}(x)$$
反函数求导法则
[(1)设y=f(x)可导且f^{'}(x)
eq0，又x=phi(y)为其反函数，则x=phi(y)可导，且\
phi^{'}(y)=frac{1}{f^{'}(x)}
\
设y=f(x)二阶可导且f^{'}(x)
eq0，又x=phi(y)为其反函数，则x=phi(y)二阶可导，且\
phi^{''}(y)=-frac{f^{''}(x)}{f^{'3}(x)}
]