Question

If$y=b\cos \log {\left(\frac{x}{n}\right)}^n$, then $\frac{dy}{dx}$ is equal to ?

• A. $-nb\sin \log {\left(\frac{x}{n}\right)}^n$
• B. $nb\sin \log {\left(\frac{x}{n}\right)}^n$
• C. $\left(\frac{-nb}{x}\right)\sin \log {\left(\frac{x}{n}\right)}^n$
• D. none of these

Answer 1

The correct option is C

$\left(\frac{-nb}{x}\right)\sin \log {\left(\frac{x}{n}\right)}^n$

Explanation for the correct answer:

To find the value of $\frac{dy}{dx}$ :

Given,

$y=b\cos \log {\left(\frac{x}{n}\right)}^n$

Evaluate the given function

$y=b\cos n\log \left(\frac{x}{n}\right)$ $\left[\because log{x}^n=nlogx\right]$

Now differentiate the obtained function with respect to $x$,

$\begin{array}{rcl}\frac{dy}{dx}& =& -b\sin n\log \frac{x}{n}\times n\left(\frac{1}{\left(x/n\right)}\right)\times \frac{1}{n}\\ & \Rightarrow & \frac{dy}{dx}=\left(-\frac{bn}{x}\right)\sin n\log \frac{x}{n}\\ & \Rightarrow & \frac{dy}{dx}=\left(-\frac{bn}{x}\right)\sin \log {\left(\frac{x}{n}\right)}^n\end{array}$

Hence, the correct option is C.