求下列函数的导数
1. $y=e^x(x\cos x-\sin x)$
2. $y=\dfrac{x\sin x}{1+\tan x}$
3. $s(t)=a\sin^2(\omega t+\varphi)$
1. $y=e^x(x\cos x-\sin x)$
2. $y=\dfrac{x\sin x}{1+\tan x}$
3. $s(t)=a\sin^2(\omega t+\varphi)$
1
(1) $y(x)=e^x(x\cos x-\sin x)$, 则
\[
\begin{split}
y'&=e^x(x\cos x-\sin x)+e^x(\cos x-x\sin x-\cos x)\\
&=e^x(x\cos x-x\sin x-\sin x)
\end{split}
\]
(2)
\[
y(x)=\frac{x\sin x}{1+\tan x}
\]
于是
\[
\begin{split}
y'&=\frac{(\sin x+x\cos x)(1+\tan x)-(x\sin x)\cdot\sec^2 x}{(1+\tan x)^2}\\
&=\frac{\sin x+\sin x\tan x+x\cos x+x\sin x-x\sin x\sec^2 x}{(1+\tan x)^2}\\
&=\frac{\sin x+\sin x\tan x+x\cos x-x\sin x\tan^2 x}{(1+\tan x)^2}
\end{split}
\]
(3) $s(t)=a\sin^2(\omega t+\varphi)$, 则
\[
\begin{split}
s'(t)&=a\cdot 2\sin(\omega t+\varphi)\cdot\cos(\omega t+\varphi)\cdot\omega\\
&=a\omega\cdot\sin\bigl(2(\omega t+\varphi)\bigr).
\end{split}
\]
或者
\[
s(t)=\frac{a}{2}\Bigl(1-\cos\bigl(2(\omega t+\varphi)\bigr)\Bigr)
\]
于是,
\[
\begin{split}
s'(t)&=-\frac{a}{2}\cdot(-1)\sin\bigl(2(\omega t+\varphi)\bigr)\cdot 2\omega\\
&=a\omega\sin\bigl(2(\omega t+\varphi)\bigr).
\end{split}
\]