Question

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

• A. $\frac{x^{5}\,cos\,x}{sin^{2}\,x}$
• B. $\frac{1}{sin\,x}-\frac{x^{5}\,cos\,x}{sin^{2}\,x}$
• C. $\frac{x}{sin^{2}\,x}$
• D. None of these

Answer 1

Answer: (D)

Let $f\left(x\right)=\frac{x^{5}-cos\,x}{sin\,x}$
$f'\left(x\right)=\frac{\left(x^{5}-cos\,x\right)'sin\,x-\left(x^{5}-cos\,x\right)\left(sin\,x\right)'}{\left(sin\,x\right)^{2}}$
$\{$using quotient rule$\}$
$=\frac{\left(5x^{4}+sin\,x\right)sin\,x-\left(x^{5}-cos\,x\right)cos\,x}{sin^{2}\,x}$
$=\frac{-x^{5}\,cos\,x+5x^{4}\,sin\,x+1}{sin^{2}\,x}$