Question
Differentiate the following functions with respect to x:
$\frac{\text{x}^5-\cos\text{x}}{\sin\text{x}}$
$\frac{\text{x}^5-\cos\text{x}}{\sin\text{x}}$
$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}^5-\cos\text{x}}{\sin\text{x}}\Big)$
Using quotient rule, we get
$\frac{(\sin\text{x})\frac{\text{d}}{\text{dx}}(\text{x}^5-\cos\text{x})-(\text{x}^5-\cos\text{x})\frac{\text{d}}{\text{dx}}(\sin\text{x})}{(\sin^2\text{x})}$
$=\frac{\sin\text{x}(\text{5x}^4\sin\text{x})-(\text{x}^5-\cos\text{x})\cos\text{x}}{(\sin^2\text{x})}$
$=\frac{\text{5x}^4\sin\text{x}+\sin^2\text{x}-\text{x}^5\cos\text{x}+\cos^2\text{x}}{(\sin^2\text{x})}\ (\because\sin^2\text{x}+\cos^2\text{x}=1)$
$=\frac{-\text{x}^5\cos\text{x}+\text{5x}^4\sin\text{x}+1}{(\sin^2\text{x})}$
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Focus is (2, 2) directrix is $\text{x+y}=\text{9}$ and eccentricity = 2