Question

The rate of change of $\sqrt{(x^2+16)}$ with respect to $\left[\frac{x}{x-1}\right]$ at$x=3$is

• A. $2$
• B. $\frac{11}{5}$
• C. $-\frac{12}{5}$
• D. $–3$

Answer 1

The correct option is C

$-\frac{12}{5}$

Step 1: Find the derivative of the given functions

Let the function $\sqrt{\left(x^2+16\right)}=y$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{d\sqrt{\left(x^2+16\right)}}{dx} \\ \Rightarrow \frac{dy}{dx}=\frac{1}{2}\left(\frac{2x}{\sqrt{\left(x^2+16\right)}}\right) \\ \Rightarrow \frac{dy}{dx}=\frac{x}{\sqrt{\left(x^2+16\right)}}.....\left(1\right) \end{gathered}$$

Now, let the function $\left[\frac{x}{x-1}\right]=z$

$\Rightarrow \frac{dz}{dx}=\frac{d\left[{\displaystyle \frac{x}{x-1}}\right]}{dx}$

$$\begin{gathered} \Rightarrow \frac{dz}{dx}=\frac{(x-1)-x}{{\left(x-1\right)}^2} \\ \Rightarrow \frac{dz}{dx}=\frac{-1}{{\left(x-1\right)}^2}.....\left(2\right) \end{gathered}$$

Step 2: Find the value of ${\left(\frac{dy}{dz}\right)}_{x=3}$

Now, dividing $\left(1\right)$ with $\left(2\right)$

$$\begin{gathered} \Rightarrow \frac{{\displaystyle \frac{dy}{dx}}}{{\displaystyle \frac{dz}{dx}}}=\frac{{\displaystyle \frac{x}{\sqrt{\left(x^2+16\right)}}}}{{\displaystyle \frac{-1}{{\left(x-1\right)}^2}}} \\ \Rightarrow \frac{dy}{dz}=\frac{-x{\left(x-1\right)}^2}{\sqrt{\left(x^2+16\right)}} \end{gathered}$$

Putting $x=3$

$$\begin{gathered} \Rightarrow \frac{dy}{dz}=\frac{-3\times {\left(3-1\right)}^2}{\sqrt{9+16}} \\ \Rightarrow \frac{dy}{dz}=\frac{-12}{5} \end{gathered}$$

Hence, the correct option is $C$, $-\frac{12}{5}$.