三角函数的n阶导数 设y=(sinx)^4+(cosx)^4,求y^(n) 意思是求n阶导y=((sinx)^2+(cosx)^2)^2-((2(sinx)^2)(cosx)^2)=1-1/2(sin2x)^2=3/4+1/4(cos4x),y的n阶导=(3/4+1/4(cos4x))^(n)=1/4*(4^n) cos(4x+npia/2)=4^(n-1)cos(4x+npia/2)问题在y的n阶
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![三角函数的n阶导数 设y=(sinx)^4+(cosx)^4,求y^(n) 意思是求n阶导y=((sinx)^2+(cosx)^2)^2-((2(sinx)^2)(cosx)^2)=1-1/2(sin2x)^2=3/4+1/4(cos4x),y的n阶导=(3/4+1/4(cos4x))^(n)=1/4*(4^n) cos(4x+npia/2)=4^(n-1)cos(4x+npia/2)问题在y的n阶](/uploads/image/z/5257248-24-8.jpg?t=%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E7%9A%84n%E9%98%B6%E5%AF%BC%E6%95%B0+%E8%AE%BEy%3D%28sinx%29%5E4%2B%28cosx%29%5E4%2C%E6%B1%82y%5E%28n%29+%E6%84%8F%E6%80%9D%E6%98%AF%E6%B1%82n%E9%98%B6%E5%AF%BCy%3D%28%28sinx%29%5E2%2B%28cosx%29%5E2%29%5E2-%28%282%28sinx%29%5E2%29%28cosx%29%5E2%29%3D1-1%2F2%28sin2x%29%5E2%3D3%2F4%2B1%2F4%28cos4x%29%2Cy%E7%9A%84n%E9%98%B6%E5%AF%BC%3D%283%2F4%2B1%2F4%28cos4x%29%29%5E%28n%29%3D1%2F4%2A%284%5En%29+cos%284x%2Bnpia%2F2%29%3D4%5E%28n-1%29cos%284x%2Bnpia%2F2%29%E9%97%AE%E9%A2%98%E5%9C%A8y%E7%9A%84n%E9%98%B6)
三角函数的n阶导数 设y=(sinx)^4+(cosx)^4,求y^(n) 意思是求n阶导y=((sinx)^2+(cosx)^2)^2-((2(sinx)^2)(cosx)^2)=1-1/2(sin2x)^2=3/4+1/4(cos4x),y的n阶导=(3/4+1/4(cos4x))^(n)=1/4*(4^n) cos(4x+npia/2)=4^(n-1)cos(4x+npia/2)问题在y的n阶
三角函数的n阶导数 设y=(sinx)^4+(cosx)^4,求y^(n) 意思是求n阶导
y=((sinx)^2+(cosx)^2)^2-((2(sinx)^2)(cosx)^2)=1-1/2(sin2x)^2=3/4+1/4(cos4x),
y的n阶导=(3/4+1/4(cos4x))^(n)=1/4*(4^n) cos(4x+npia/2)=4^(n-1)cos(4x+npia/2)
问题在y的n阶导这一行.这一式子看不懂.尤其是从上一步到这一步.本人基础不好.越详细越好,
三角函数的n阶导数 设y=(sinx)^4+(cosx)^4,求y^(n) 意思是求n阶导y=((sinx)^2+(cosx)^2)^2-((2(sinx)^2)(cosx)^2)=1-1/2(sin2x)^2=3/4+1/4(cos4x),y的n阶导=(3/4+1/4(cos4x))^(n)=1/4*(4^n) cos(4x+npia/2)=4^(n-1)cos(4x+npia/2)问题在y的n阶
关键是cosx的n阶导数
cos'x=-sinx=cos(π/2+x)
cos''x=-cosx=cos(π+x)
cos'''x=sinx=cos(3π/2+x)
cosx的4阶导数=cosx=cos(2π+x)
因此cosx的n阶导数=cos(4x+nπ/2)
(3/4+1/4(cos4x)的n阶导=1/4(cos4x)的n阶导=4^(n-1)cos(4x+nπ/2)