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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91影视

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.

Most popular questions from this chapter

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91影视

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

Translational Dynamics

Magnetism and Electromagnetic Induction

Electrostatics

Further Mechanics and Thermal Physics

Dynamics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

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