Question

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

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = \ln x$ and $v=\cos x$.

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

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

Step3: Calculate $f'(x)$

$f'(x)=u'v + uv'=\frac{1}{x}\cdot\cos x+\ln x\cdot(-\sin x)=\frac{\cos x}{x}-\ln x\sin x$.

Answer:

C. $\frac{\cos x}{x}-\ln x\sin x$