Differentiate the following functions with respect to x: a+b c+d - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\text{a}+\text{b}\sin\text{x}}{\text{c}+\text{d}\cos\text{x}}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{a}+\text{b}\sin\text{x}}{\text{c}+\text{d}\cos\text{x}}\Big)$
Using quotient rule, we get
$\frac{(\text{c}+\text{d}\cos\text{x})\frac{\text{d}}{\text{dx}}(\text{a}+\text{b}\sin\text{x})-(\text{a}+\text{b}\sin\text{x})\frac{\text{d}}{\text{dx}}(\text{c}+\text{d}\cos\text{x})}{(\text{c}+\text{d}\cos\text{x})^2}$
$=\frac{(\text{c}+\text{d}\cos\text{x})(\text{b}\cos\text{x})-(\text{a}+\text{b}\sin\text{x})(-\text{d}\sin\text{x})}{(\text{c}+\text{d}\cos\text{x})^2}$
$=\frac{\text{bc}\cos\text{x}+\text{bd}\cos^2\text{x}+\text{ad}\sin\text{x}+\text{bd}\sin^2\text{x}}{(\text{c}+\text{d}\cos\text{x})^2}$
$=\frac{\text{bc}\cos\text{x}+\text{ad}\sin\text{x}+\text{bd}(\sin^2\text{x}+\cos^2\text{x})}{(\text{c}+\text{d}\cos\text{x})^2}$
$=\frac{\text{bc}\cos\text{x}+\text{ad}\sin\text{x}+\text{bd}}{(\text{c}+\text{d}\cos\text{x})^2}$

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