求y=sin^nx cos^nx的导数 y=sin^n(x)cos nx 导数

\u6c42\u4e0b\u5217\u51fd\u6570\u5bfc\u6570 y=sinnx•sin^nx(n\u4e3a\u5e38\u6570)

\u5b9e\u9645\u4e0a\u518d\u8fdb\u884c\u51e0\u6b65\u7684\u5316\u7b80\u5373\u53ef

y=sinnx *(sinx)^n
\u90a3\u4e48\u6c42\u5bfc\u5f97\u5230
y'=(sinnx)' *(sinx)^n +sinnx *[(sinx)^n]'
=n *cosnx *(sinx)^n +sinnx * n *(sinx)^n-1 *cosx
\u63d0\u53d6\u51fan *(sinx)^n-1 *cosx
\u5f97\u5230y'=n *(sinx)^n-1 *(sinx *cosnx +sinnx *cosx)
\u800c\u7531\u516c\u5f0fsin(a+b)=sina *cosb +sinb *cosa\u5f97\u5230
sinx *cosnx +sinnx *cosx =sin(nx+x)=sin(n+1) x
\u518d\u8fdb\u884c\u4ee3\u5165\u5c31\u5f97\u5230
y' =n *(sinx)^n-1 *(sinx *cosnx +sinnx *cosx)
=n *(sinx)^n-1 *sin(n+1)x
\u5c31\u662f\u4f60\u8981\u7684\u7b54\u6848

[sin^n(x)]'=nsin^(n-1)(x)cosx
[cosnx]'=-nsinnx
y'=[sin^n(x)]'cos nx +[cosnx]'sin^n(x)
=nsin^(n-1)(x)cosxcos nx-nsinnxsin^n(x)
=nsin^(n-1)(x)(coscosnx-sinnxsinx)
=nsin^(n-1)(x)cos(n+1)x
cos(a+b)=cosacosb-sinasinb \u516c\u5f0f\u554a\uff0c\u54e5\u54e5

y=sin^nx cos^nx
y′=nsin^(n-1)xcosxcos^nx+ncos^(n-1)x(-sinx)sin^nx
=nsin^(n-1)xcos^(n-1)x(cos²x-sin²x)
=nsin^(n-1)xcos^(n-1)cos(2x)

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