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Question Number 9491 by Raja Naik last updated on 10/Dec/16
find the value of (d/dx)(((acos^3 x)/(3sinx^2  )))  and it is possible or not
$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{acos}^{\mathrm{3}} \mathrm{x}}{\mathrm{3sinx}^{\mathrm{2}} \:}\right) \\ $$$$\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{or}\:\mathrm{not} \\ $$
Answered by mrW last updated on 11/Dec/16
y=((acos^3 x)/(3sin x^2 ))  (dy/dx)=(1/3)(acos^3 x)(−(1/(sin^2 x^2 )))(cos x^2 )2+(1/(3sin x^2 ))(3acos^2 x)(−(1/( (√(1−x^2 )))))  =−((acos^2 x)/(3sin x^2 ))[((2(acos x)(cos x^2 ))/(sin x^2 ))+(3/( (√(1−x^2 ))))]
$$\mathrm{y}=\frac{\mathrm{acos}^{\mathrm{3}} \mathrm{x}}{\mathrm{3sin}\:\mathrm{x}^{\mathrm{2}} } \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{acos}^{\mathrm{3}} \mathrm{x}\right)\left(−\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} }\right)\left(\mathrm{cos}\:\mathrm{x}^{\mathrm{2}} \right)\mathrm{2}+\frac{\mathrm{1}}{\mathrm{3sin}\:\mathrm{x}^{\mathrm{2}} }\left(\mathrm{3acos}^{\mathrm{2}} \mathrm{x}\right)\left(−\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\right) \\ $$$$=−\frac{\mathrm{acos}^{\mathrm{2}} \mathrm{x}}{\mathrm{3sin}\:\mathrm{x}^{\mathrm{2}} }\left[\frac{\mathrm{2}\left(\mathrm{acos}\:\mathrm{x}\right)\left(\mathrm{cos}\:\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{sin}\:\mathrm{x}^{\mathrm{2}} }+\frac{\mathrm{3}}{\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\right] \\ $$

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