Chapter 4: Q1P (page 203)

If $p v^{a} = c$ (where $a$ and $c$ are constants) find $\frac{dv}{dp}$ and $\frac{d^{2} v}{d^{2} p}$ .

Short Answer

Expert verified

Derivative of the equation ${pv}^{a} = C$ is $\frac{dv}{dp} = - \frac{v}{ap}$ .

The second derivative of the equation ${pv}^{a} = C$ is $\frac{d^{2} v}{d^{2} p} = \frac{v (1 + a)}{a^{2} p}$ .

Step by step solution

Explanation of Solution

The provided equation is ${pv}^{a} = C$ .

Step 2: Implicit Differentiation

In implicit differentiation, we differentiate each side of an equation with two variables (usually $x$ androle="math" localid="1658828364260" $y$ ) by treating one of the variables as a function of the other.

Calculation

Consider the equation below:

${pv}^{a} = C$

Here a and C are constants.

The provided equation ${pv}^{a} = C$ is written as follows:

$v^{a} = \frac{c}{p}$

Take precautions on all sides $(In)$ .

$In (v^{a}) = In (\frac{c}{p})$

Implies that,

$aln (v) = In (c) - In (p)$

Consider the derivative with respect to p,

$\frac{a}{v} \frac{dv}{dp} = 0 - \frac{1}{p}$

Since,

$\frac{dv}{dp} = - \frac{v}{ap}$

Consider the first derivative of the equation below:

${pv}^{a} = C$ with respect to $p$ .

Make a new distinction with reference to $p$ .

$\frac{d^{2} v}{d^{2} p} = - \frac{1}{a} [\frac{p \frac{dv}{dp} - v (1)}{p^{2}}] = - \frac{1}{ap} [\frac{p (- \frac{v}{ap}) - v}{p^{2}}] = - \frac{1}{ap} [\frac{- \frac{v}{ap} - v}{p^{2}}] = \frac{v (1 + a)}{a^{2} p^{2}}$

Hence,

$\frac{d^{2} v}{d^{2} p} = \frac{v (1 + a)}{a^{2} p^{2}}$ .

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If $p v^{a} = c$ (where $a$ and $c$ are constants) find $\frac{dv}{dp}$ and $\frac{d^{2} v}{d^{2} p}$ .

Short Answer

Step by step solution

Explanation of Solution

Step 2: Implicit Differentiation

Calculation

One App. One Place for Learning.

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Ifpva=c (where aandc are constants) finddvdp andd2vd2p .

Short Answer

Step by step solution

Explanation of Solution

Step 2: Implicit Differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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If $p v^{a} = c$ (where $a$ and $c$ are constants) find $\frac{dv}{dp}$ and $\frac{d^{2} v}{d^{2} p}$ .