Question

if (f(x)=\frac{cos x}{ln x}), then (f(x)=)
a (\frac{xsin xln x - cos x}{x(ln x)^2})
b (\frac{-xsin xln x - cos x}{x(ln x)^2})
c (-xsin x)
d (\frac{-xsin xln x+cos x}{x(ln x)^2})

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = \cos x$, $u'=-\sin x$, $v=\ln x$, and $v'=\frac{1}{x}$.

Step2: Substitute values into quotient - rule

$f'(x)=\frac{(-\sin x)\cdot\ln x-\cos x\cdot\frac{1}{x}}{(\ln x)^{2}}=\frac{-x\sin x\ln x - \cos x}{x(\ln x)^{2}}$

Answer:

B. $\frac{-x\sin x\ln x - \cos x}{x(\ln x)^{2}}$