Question

Find the derivative of $(x^2+1)\cos x$.

Answer 1

Let $f\left(x\right)=(x^2+1)\cos x$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(\right(x^2+1\left)\cos x\right)$
$\Rightarrow f^{\prime }\left(x\right)=\left(\cos x\right)\frac{d}{dx}(x^2+1)+(x^2+1)\frac{d}{dx}\left(\cos x\right)$
$\Rightarrow f^{\prime }\left(x\right)=\cos x(2x+0)+(x^2+1)(-\sin x)$
$\Rightarrow f^{\prime }\left(x\right)=2x\cos x-\sin x(x^2+1)$ $\Rightarrow f^{\prime }\left(x\right)=2x\cos x-x^2\sin x-\sin x$
$\therefore f^{\prime }\left(x\right)=-x^2\sin x+2x\cos x-\sin x$