掌握高级导数技巧：AM课程中的高阶导数法则及常见基本函数的高阶导数公式详解
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}x^{\alpha}=x^{\alpha-n}\prod_{k=0}^{n-1}(\alpha-k)
dxndnxα=xα−n∏k=0n−1(α−k)
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}\frac{1}{x}=(-1)^{n}\frac{n!}{x^{n+1}}
dxndnx1=(−1)nxn+1n!
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}\ln x=(-1)^{n-1}\frac{(n-1)!}{x^n}
dxndnlnx=(−1)n−1xn(n−1)!;
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}\ln(1+x)
dxndnln(1+x)=
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(-1)^{n-1}\frac{(n-1)!}{(1+x)^{n}}
(−1)n−1(1+x)n(n−1)!
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}e^x=e^x
dxndnex=ex
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\frac{{\rm{d}}^n}{{\rm{d}}x^n} a^x=a^x \cdot \ln^n a
dxndnax=ax⋅lnna
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}\sin \left(kx+b\right)=k^n\sin \left(kx+b+\frac{n\pi}{2}\right)
dxndnsin(kx+b)=knsin(kx+b+2nπ)
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\frac{{\rm{d}}^n}{{\rm{d}}x^n}\cos \left(kx+b\right)=k^n\cos \left(kx+b+\frac{n\pi}{2}\right)
dxndncos(kx+b)=kncos(kx+b+2nπ)