隐函数x⁴(7x²+3y²)=(5x²-14y²)²的导数
主要内容:
本文通过隐函数求导规则,介绍计算隐函数x⁴(7x²+3y²)=(5x²-14y²)²的导数
的主要步骤。
主要步骤:
解:对方程x⁴(7x²+3y²)=(5x²-14y²)²两边同时对x求导得:
4x³(7x²+3y²)+x⁴(14x+6yy')=2(5x²-14y²)(10x-28yy'),
方程两边同时除以2,得:
2x³(7x²+3y²)+x⁴(7x+3yy')=2(5x²-14y²)(5x-14yy'),
2x³(7x²+3y²)+7x⁵+3yx⁴y'=10(5x²-14y²)x-28y(5x²-14y²)y',
移动含有y'的项到等号左边,有:
[3yx⁴+28y(5x²-14y²)]y'=10(5x²-14y²)x-2x³(7x²+3y²)-7x⁵,
方程两边提取公因数,化简得:
y[3x⁴+28 (5x²-14y²)]y'=x[10(5x²-14y²)-2x²(7x²+3y²)-7x⁴],
化简得:
y'=x[10(5x²-14y²)-2x²(7x²+3y²)-7x⁴]/{y[3x⁴+28(5x²-14y²)]}.
=x(50x²-140y²-21x⁴-6x²y²)/{y[3x⁴+28(5x²-14y²)]}.
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