Question

6 mark for review 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: Recall quotient - rule

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

Step2: Find $u'$ and $v'$

The derivative of $u=\cos x$ is $u'=-\sin x$, and the derivative of $v = \ln x$ is $v'=\frac{1}{x}$.

Step3: Apply the 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. $\frac{-x\sin x\ln x - \cos x}{x(\ln x)^{2}}$