设函数y=f(x)是严格单调的三阶可导函数,而且f'(x)≠0,求(f^-1)^(3)(y)(即f(x)的反函数的三阶导数).好像是数学分析第一册习题四的最后一题.只有答案((f^-1)^(3)(y)={-f'''(x)/[f'(x)]^4}+{3(f''(x))^2/[f'(x)]
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![设函数y=f(x)是严格单调的三阶可导函数,而且f'(x)≠0,求(f^-1)^(3)(y)(即f(x)的反函数的三阶导数).好像是数学分析第一册习题四的最后一题.只有答案((f^-1)^(3)(y)={-f'''(x)/[f'(x)]^4}+{3(f''(x))^2/[f'(x)]](/uploads/image/z/2622882-66-2.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0y%3Df%28x%29%E6%98%AF%E4%B8%A5%E6%A0%BC%E5%8D%95%E8%B0%83%E7%9A%84%E4%B8%89%E9%98%B6%E5%8F%AF%E5%AF%BC%E5%87%BD%E6%95%B0%2C%E8%80%8C%E4%B8%94f%27%28x%29%E2%89%A00%2C%E6%B1%82%28f%5E-1%29%5E%283%29%28y%29%EF%BC%88%E5%8D%B3f%28x%29%E7%9A%84%E5%8F%8D%E5%87%BD%E6%95%B0%E7%9A%84%E4%B8%89%E9%98%B6%E5%AF%BC%E6%95%B0%EF%BC%89.%E5%A5%BD%E5%83%8F%E6%98%AF%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90%E7%AC%AC%E4%B8%80%E5%86%8C%E4%B9%A0%E9%A2%98%E5%9B%9B%E7%9A%84%E6%9C%80%E5%90%8E%E4%B8%80%E9%A2%98.%E5%8F%AA%E6%9C%89%E7%AD%94%E6%A1%88%EF%BC%88%28f%5E-1%29%5E%283%29%28y%29%3D%7B-f%27%27%27%28x%29%2F%5Bf%27%28x%29%5D%5E4%7D%2B%7B3%28f%27%27%28x%29%29%5E2%2F%5Bf%27%28x%29%5D)
设函数y=f(x)是严格单调的三阶可导函数,而且f'(x)≠0,求(f^-1)^(3)(y)(即f(x)的反函数的三阶导数).好像是数学分析第一册习题四的最后一题.只有答案((f^-1)^(3)(y)={-f'''(x)/[f'(x)]^4}+{3(f''(x))^2/[f'(x)]
设函数y=f(x)是严格单调的三阶可导函数,而且f'(x)≠0,求(f^-1)^(3)(y)(即f(x)的反函数的三阶导数).
好像是数学分析第一册习题四的最后一题.只有答案((f^-1)^(3)(y)={-f'''(x)/[f'(x)]^4}+{3(f''(x))^2/[f'(x)]^5}),
设函数y=f(x)是严格单调的三阶可导函数,而且f'(x)≠0,求(f^-1)^(3)(y)(即f(x)的反函数的三阶导数).好像是数学分析第一册习题四的最后一题.只有答案((f^-1)^(3)(y)={-f'''(x)/[f'(x)]^4}+{3(f''(x))^2/[f'(x)]
首先:(f^-1)^(1)(y)=1/f' (这没问题吧,约定f=f(x))
得: (f^-1)^(2)(y)=d((f^-1)^(1)(y))/dy
=d((f^-1)^(1)(y))/dx*1/f'
=1/f'*d(1/f')/dx
=1/f'*(-f''/(f')^2)
=-f''/(f')^3;
得: (f^-1)^(3)(y)=d((f^-1)^(2)(y))/dy
=d((f^-1)^(2)(y))/dx*1/f'
=1/f'*d(-f''/(f')^3)dx
={-f'''(x)/[f'(x)]^4}+{3(f''(x))^2/[f'(x)]^5}