Arely Briggs

Answered

2022-01-24

Derivative of a product of trig functions: $3\mathrm{sin}\left(x\right)\mathrm{cot}\left(x\right)$
$\frac{-3\mathrm{cos}x}{{\mathrm{sin}}^{2}x}$

Answer & Explanation

Maritza Mccall

Expert

2022-01-25Added 17 answers

You need to use the product rule:
$\left(3\mathrm{sin}x\mathrm{cot}x\right)\prime =$
$3\left(\mathrm{sin}x\right)\prime \mathrm{cot}x+3\mathrm{sin}x\left(\mathrm{cot}x\right)\prime =$
$3\mathrm{cos}x\mathrm{cot}x+3\mathrm{sin}x\left(-{\mathrm{csc}}^{2}x\right)=$
$3\mathrm{cos}x\mathrm{cot}x-3\mathrm{sin}x{\mathrm{csc}}^{2}x=$
$3\mathrm{cos}x\mathrm{cos}x\mathrm{sin}x-3\mathrm{sin}x\frac{1}{{\mathrm{sin}}^{2}x}=$
$3\frac{{\mathrm{cos}}^{2}x}{\mathrm{sin}x}-3\frac{1}{\mathrm{sin}x}=$
$3\frac{1}{\mathrm{sin}x}\left({\mathrm{cos}}^{2}x-1\right)=$
$3\frac{1}{\mathrm{sin}x}\left(1-{\mathrm{sin}}^{2}x-1\right)=-3\mathrm{sin}x$
A much simpler way to do this, as was mentioned in the comments section, is to notice that $3\mathrm{sin}x\mathrm{cot}x=3\mathrm{sin}x\frac{\mathrm{cos}x}{\mathrm{sin}x}=3\mathrm{cos}x$ which is trivial to differentiate:
$\left(3\mathrm{cos}x\right)\prime =3\left(\mathrm{cos}x\right)\prime =3\left(-\mathrm{sin}x\right)=-3\mathrm{sin}x$

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