Question

If $ pv=81 $ , then $ \dfrac{dp}{dv} $ is, at $ v=9 $ , equal to
(a) 1
(b) $ -1 $
(c) 2
(d) None of these

Answer 1

Hint: We will find the value of $ p $ in terms of $ v $ using the given equation. Then we will differentiate $ p $ with respect to $ v $ . We will substitute the value of $ v $ , which is given, in the obtained equation. After simplifying this equation,, we will get the value of $ \dfrac{dp}{dv} $ . We will use the differentiation of $ \dfrac{1}{x} $ , which is $ -\dfrac{1}{{{x}^{2}}} $ . We will also use the formula of differentiation $ \dfrac{d}{dx}\left( cy \right)=c\dfrac{dy}{dx} $ where $ c $ is a constant.

Complete step by step answer:
We are given that $ pv=81 $ . Dividing both sides of this equation by $ v $ , we get the following
 $ p=\dfrac{81}{v} $
Now, we will differentiate both sides of the above equation with respect to $ v $ , as follows,
 $ \dfrac{dp}{dv}=\dfrac{d}{dv}\left( \dfrac{81}{v} \right) $
But we know that $ \dfrac{d}{dx}\left( cy \right)=c\dfrac{dy}{dx} $ where $ c $ is a constant. Therefore, we have the following
 $ \dfrac{d}{dv}\left( \dfrac{81}{v} \right)=81\cdot \dfrac{d}{dv}\left( \dfrac{1}{v} \right) $
We know the differentiation of $ \dfrac{1}{x} $ is $ \dfrac{d}{dx}\left( \dfrac{1}{x} \right)=-\dfrac{1}{{{x}^{2}}} $ . Using this formula, we have the following equation,
 $ \dfrac{dp}{dv}=81\cdot -\dfrac{1}{{{v}^{2}}} $
We have to find the value of $ \dfrac{dp}{dv} $ at $ v=9 $ . Substituting the value $ v=9 $ in the above equation, we get
 $ \begin{align}
  & \dfrac{dp}{dv}=81\cdot -\dfrac{1}{{{9}^{2}}} \\
 & \Rightarrow \dfrac{dp}{dv}=81\cdot -\dfrac{1}{81} \\
 & \therefore \dfrac{dp}{dv}=-1 \\
\end{align} $
So, we get the value of $ \dfrac{dp}{dv} $ at $ v=9 $ as $ -1 $ . Hence, the correct option is (b).

Note:
It is useful to know the derivatives of standard functions for such types of questions. The aspect is to notice that we have to find the derivative of $ p $ with respect to $ v $ and we are given an equation with the relation between them. Hence, we can obtain an equation which gives us the derivative of $ p $ with respect to $ v $ in terms of $ v $ . It is beneficial to write the equations and the derivatives explicitly so that we can avoid any errors and get the required answer.