Question

question 6: find a derivative
find $f(x)$ given
$f(x)=\frac{sec(8x)}{9x}$.
$f(x)=$ symbolic expression

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = \sec(8x)$ and $v = 9x$.

Step2: Find $u'$

Using the chain - rule, if $u=\sec(8x)$, let $t = 8x$, then $\frac{du}{dt}=\sec(t)\tan(t)$ and $\frac{dt}{dx}=8$. So, $u'=\frac{d}{dx}\sec(8x)=8\sec(8x)\tan(8x)$.

Step3: Find $v'$

Since $v = 9x$, then $v'=9$.

Step4: Substitute into quotient - rule

$f'(x)=\frac{8\sec(8x)\tan(8x)\cdot9x-\sec(8x)\cdot9}{(9x)^{2}}=\frac{72x\sec(8x)\tan(8x)-9\sec(8x)}{81x^{2}}=\frac{\sec(8x)(8x\tan(8x) - 1)}{9x^{2}}$

Answer:

$\frac{\sec(8x)(8x\tan(8x)-1)}{9x^{2}}$