Differentiate the following functions. a+b c+d - Vidyadip

Question

Differentiate the following functions.
$\frac{\text{a}+\text{b}\sin\text{x}}{\text{c}+\text{d}\cos\text{x}}$

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Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{a}+\text{b}\sin\text{x}}{\text{c}+\text{d}\cos\text{x}}\Big)$
$=\frac{(\text{c}+\text{d}\cos\text{x}).\frac{\text{d}}{\text{dx}}(\text{a}+\text{b}\sin\text{x})-(\text{a}+\text{b}\sin\text{x})\frac{\text{d}}{\text{dx}}(\text{c}+\text{d}\cos\text{x})}{(\text{c}+\text{d}\cos\text{x})^{2}}$
$=\frac{\text{cb}\cos\text{x}+\text{bd}\cos^{2}\text{x}+\text{ad}\sin\text{x}+\text{bd}\sin^{2}\text{x}}{(\text{c}+\text{d}\cos\text{x})^{2}}$
$=\frac{\text{cb}\cos\text{x}+\text{ad}\sin\text{x}+\text{bd}(\cos^{2}\text{x}+\sin^{2}\text{x})}{(\text{c}+\text{d}\cos\text{x}^{2})}$
$=\frac{\text{cb}\cos\text{x}+\text{ad}\sin\text{x}+\text{bd}}{(\text{c}+\text{d}\cos\text{x})^{2}}$

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