设f(x)可导,且f`(x)=sin^2[sin(x+1)].f(0)=4.求f(x)的反函数当自变量取4时的导数值
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![设f(x)可导,且f`(x)=sin^2[sin(x+1)].f(0)=4.求f(x)的反函数当自变量取4时的导数值](/uploads/image/z/8457959-47-9.jpg?t=%E8%AE%BEf%28x%29%E5%8F%AF%E5%AF%BC%2C%E4%B8%94f%60%28x%29%3Dsin%5E2%5Bsin%28x%2B1%29%5D.f%280%29%3D4.%E6%B1%82f%28x%29%E7%9A%84%E5%8F%8D%E5%87%BD%E6%95%B0%E5%BD%93%E8%87%AA%E5%8F%98%E9%87%8F%E5%8F%964%E6%97%B6%E7%9A%84%E5%AF%BC%E6%95%B0%E5%80%BC)
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设f(x)可导,且f`(x)=sin^2[sin(x+1)].f(0)=4.求f(x)的反函数当自变量取4时的导数值
设f(x)可导,且f`(x)=sin^2[sin(x+1)].f(0)=4.求f(x)的反函数当自变量取4时的导数值
设f(x)可导,且f`(x)=sin^2[sin(x+1)].f(0)=4.求f(x)的反函数当自变量取4时的导数值
f`(x)=sin^2[sin(x+1)].
f'(0)=sin^2(sin1)
f(0)=4
f(x)图像过(0,4)点,设在(0,4)点处曲线f(x)的切线l1
倾斜角为α,那么tanα=f'(0)
f(x)与其反函数g(x)图像关于直线y=x对称
那么g(4)=0,设g(x)在(4,0)点处的切线l2倾斜角为β
那么l1,l2关于直线y=x对称,
∴α+β=π/2,
∴g'(4)=tanβ=1/tanα=1/f'(0)=1/sin^2(sin1)
即求f(x)的反函数当自变量取4时的导数值为
1/[sin^2(sin1)]
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