Question
Differentiate the following functions with respect to x:
$\frac{\text{a}+\sin\text{x}}{1+\text{a}\sin\text{x}}$
$\frac{\text{a}+\sin\text{x}}{1+\text{a}\sin\text{x}}$
$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{a}+\sin\text{x}}{1+\text{a}\sin\text{x}}\Big)$
Using quotient rule, we get
$\frac{(1+\text{a}\sin\text{x})\frac{\text{d}}{\text{dx}}(\text{a}+\sin\text{x})-(\text{a}+\sin\text{x})\frac{\text{d}}{\text{dx}}(1+\text{a}\sin\text{x})}{(1+\text{a}\sin\text{x})^2}$
$=\frac{(1+\text{a}\sin\text{x})\cos\text{x}-(\text{a}+\sin\text{x})\text{a}\cos\text{x}}{(1+\text{a}\sin\text{x})^2}$
$=\frac{\cos\text{x}+\text{a}\sin\text{x}\cos\text{x}-\text{a}^2\cos\text{x}-\text{a}\sin\text{x}\cos\text{x}}{(1+\text{a}\sin\text{x})^2}$
$=\frac{(1-\text{a}^2)\cos\text{x}}{(1+\text{a}\sin\text{x})^2}$
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