differentiate. f(θ) = sec θ / 7 + sec θ f (θ) =

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = \sec\theta$, $u'=\sec\theta\tan\theta$, $v = 7+\sec\theta$, and $v'=\sec\theta\tan\theta$.

Step2: Apply quotient - rule

[ \begin{align*} f'(\theta)&=\frac{\sec\theta\tan\theta(7 + \sec\theta)-\sec\theta(\sec\theta\tan\theta)}{(7+\sec\theta)^{2}}\ &=\frac{7\sec\theta\tan\theta+\sec^{2}\theta\tan\theta-\sec^{2}\theta\tan\theta}{(7 + \sec\theta)^{2}}\ &=\frac{7\sec\theta\tan\theta}{(7+\sec\theta)^{2}} \end{align*} ]

Answer:

$\frac{7\sec\theta\tan\theta}{(7+\sec\theta)^{2}}$