Question

Derivative of which function is $f^{\prime }\left(x\right)=x\sin x$?

• A. $x\sin x+\cos x$
• B. $x\cos x+\sin x$
• C. $x\sin (\frac{\pi }{2}-x)+\cos (\frac{\pi }{2}-x)$
• D. $x\cos (\frac{\pi }{2}-x)+\sin (\frac{\pi }{2}-x)$

Answer 1

Answer: (B), (C)

$x\cos x+\sin x$
$x\sin (\frac{\pi }{2}-x)+\cos (\frac{\pi }{2}-x)$

We know $\frac{d}{dx}(f.g)=\frac{df}{dx}.g+\frac{dg}{dx}.f\Rightarrow \frac{d}{dx}(x.sin(x\left)\right)=\frac{d\left(x\right)}{dx}.sin\left(x\right)+\frac{d\left(sin\right(x\left)\right)}{dx}.x\Rightarrow 1\times sin\left(x\right)+cos\left(x\right).x=x.cos\left(x\right)+sin\left(x\right)$

We know that $cos\left(x\right)=sin(\frac{\pi }{2}-x)sin\left(x\right)=cos(\frac{\pi }{2}-x)\Rightarrow x.sin((\frac{\pi }{2}-x)+cos(\frac{\pi }{2}-x)$

Both B and C are correct