Choose useful trig identities before differentiating

$d(x^2\sin x)/dx=x^2\cos x+2x\sin x=x(x\cos x+2\sin x)$, for all real $x$. All angles are in radians.

Where $\cos x\ne0$, $(1+\tan x)/\sec x=\cos x+\sin x$. Its derivative is $-\sin x+\cos x$. Keep $\cos x\ne0$: the simplified function has values at excluded inputs, but the original expression does not.

Write $(\cos 2x)^{1/2}$. Then $y^{\prime}=\frac12(\cos 2x)^{-1/2}(-\sin2x)(2)=-\sin2x/\sqrt{\cos2x}$. The source’s Teacher-filled chain steps are retained. The real function requires $\cos2x\ge0$; the finite derivative formula requires $\cos2x>0$.

$d[\cot(x^2)]/dx=-\csc^2(x^2)(2x)$ and $d[\sec^2x]/dx=2\sec x(\sec x\tan x)$. Hence $y^{\prime}=-2x\csc^2(x^2)+2\sec^2x\tan x$. Retain $\sin(x^2)\ne0$ and $\cos x\ne0$. The source writes cosec where this lesson uses the equivalent notation csc.