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Mathematics Question on limits and derivatives

Find the derivative of x5−cos xsin x\frac{x^{5}-cos\,x}{sin\,x}.

A

x5 cos xsin2 x\frac{x^{5}\,cos\,x}{sin^{2}\,x}

B

1sin x−x5 cos xsin2 x\frac{1}{sin\,x}-\frac{x^{5}\,cos\,x}{sin^{2}\,x}

C

xsin2 x\frac{x}{sin^{2}\,x}

D

None of these

Answer

None of these

Explanation

Solution

Let f(x)=x5−cos xsin xf\left(x\right)=\frac{x^{5}-cos\,x}{sin\,x} f′(x)=(x5−cos x)′sin x−(x5−cos x)(sin x)′(sin x)2f'\left(x\right)=\frac{\left(x^{5}-cos\,x\right)'sin\,x-\left(x^{5}-cos\,x\right)\left(sin\,x\right)'}{\left(sin\,x\right)^{2}} \\{using quotient rule\\} =(5x4+sin x)sin x−(x5−cos x)cos xsin2 x=\frac{\left(5x^{4}+sin\,x\right)sin\,x-\left(x^{5}-cos\,x\right)cos\,x}{sin^{2}\,x} =−x5 cos x+5x4 sin x+1sin2 x=\frac{-x^{5}\,cos\,x+5x^{4}\,sin\,x+1}{sin^{2}\,x}