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:

$\left(3\mathrm{sin}x\mathrm{cot}x\right)\prime =$

$3\left(\mathrm{sin}x\right)\prime \mathrm{cot}x+3\mathrm{sin}x\left(\mathrm{cot}x\right)\prime =$

$3\mathrm{cos}x\mathrm{cot}x+3\mathrm{sin}x(-{\mathrm{csc}}^{2}x)=$

$3\mathrm{cos}x\mathrm{cot}x-3\mathrm{sin}x{\mathrm{csc}}^{2}x=$

$3\mathrm{cos}x\mathrm{cos}x\mathrm{sin}x-3\mathrm{sin}x\frac{1}{{\mathrm{sin}}^{2}x}=$

$3\frac{{\mathrm{cos}}^{2}x}{\mathrm{sin}x}-3\frac{1}{\mathrm{sin}x}=$

$3\frac{1}{\mathrm{sin}x}({\mathrm{cos}}^{2}x-1)=$

$3\frac{1}{\mathrm{sin}x}(1-{\mathrm{sin}}^{2}x-1)=-3\mathrm{sin}x$

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:

$\left(3\mathrm{cos}x\right)\prime =3\left(\mathrm{cos}x\right)\prime =3(-\mathrm{sin}x)=-3\mathrm{sin}x$

A much simpler way to do this, as was mentioned in the comments section, is to notice that

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