Differentiate the functions with respect to x. (ax + b) (cx + d) - Vidyadip

Question

Differentiate the functions with respect to x.
$\frac{\sin(\text{ax + b)}}{\cos(\text{cx + d})}$

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Answer

$\text{Let y} = \frac{\sin(\text{ax + b)}}{\cos(\text{cx + d})}$
Using quotient rule,
$\therefore \frac{\text{dy}}{\text{dx}} = \frac{\cos(\text{cx + d})\frac{\text{d}}{\text{dx}}\sin\text(\text{ax + b})-\sin(\text{ax + b)}\frac{\text{d}}{\text{dx}}\cos(\text{cx} + \text{d})}{\cos^2(\text{cx} + \text{d})}$
$= \frac{\cos(\text{cx + d})\cos\text(\text{ax + b})\frac{\text{d}}{\text{dx}}(\text{ax + b)}-\sin(\text{ax + b)}\left\{-\sin(\text{cx + d)}\right\}\frac{\text{d}}{\text{dx}}(\text{cx} + \text{d})}{\cos^2(\text{cx} + \text{d})}$
$= \frac{\cos(\text{cx + d})\cos\text(\text{ax + b})(\text{a)}+\sin(\text{ax + b})\sin(\text{cx + d)}(\text{c})}{\cos^2(\text{cx} + \text{d})}$

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