Find the derivative of x^ 5 - x x - Vidyadip

Question

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

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Answer

Let $h(x)=\frac{x^{5}-\cos x}{\sin x}$ We use the quotient rule on this function wherever it is defined.Then,we have,
$h^{\prime}(x)=\frac{\left(x^{5}-\cos x\right)^{\prime} \sin x-\left(x^{5}-\cos x\right)(\sin x)^{\prime}}{(\sin x)^{2}}$
$=\frac{\left(5 x^{4}+\sin x\right) \sin x-\left(x^{5}-\cos x\right) \cos x}{\sin ^{2} x}$
$=\frac{-x^{5} \cos x+5 x^{4} \sin x+1}{(\sin x)^{2}}$

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