Chapter 4: Q3P (page 201)

Given $r = e^{- p^{2} - q^{2}}, p = e^{5}, q = e^{- 5},$ find $\frac{d r}{d s}$ .

Short Answer

Expert verified

The value of $\frac{d r}{d s}$ is $2 r (q^{2} - p^{2})$ .

Step by step solution

Explanation of solution

The provided expressions are $r = e^{- p^{2} - q^{2}}, p = e^{s}$ , and $q = e^{- s}$ .

Chain Rule

The number of functions in the composition affects how many differentiation steps are required when using the chain rule to get the derivative of composite functions.

Calculation

Use the below equation.

$d r = \frac{\partial r}{\partial p} d p + \frac{\partial r}{\partial p} d q$

Differentiate above equation.

$\frac{\partial r}{\partial p} = e^{- p^{2} - q^{2}} (2 p) = - 2 p e^{- p^{2} - q^{2}}$

Differentiate above equation.

$\frac{\partial r}{\partial q} = e^{- p^{2} - q^{2}} (- 2 p) = - 2 p e^{- p^{2} - q^{2}}$

Calculate the value of by use equation $d r = \frac{\partial r}{\partial p} d p + \frac{\partial r}{\partial q} d q$ .

$d r = - 2 p e^{- p^{2} - q^{2}} d p - 2 p e^{- p^{2} - q^{2}} d q$

Calculate the value of $\frac{d r}{d s}$ use equation $d p = e^{s} d s$ and $d q = - e^{- s} d s$ .

$\frac{d r}{d s} = - 2 p e^{- p^{2} - q^{2}} (e^{s}) - 2 p e^{- p^{2} - q^{2}} (- e^{- s}) = 2 e^{- p^{2} - q^{2}} (- p e^{s} + q e^{- s}) = 2 r (q^{2} - p^{2})$

Hence the value of $\frac{d r}{d s}$ is $2 r (q^{2} - p^{2})$ .

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Given $r = e^{- p^{2} - q^{2}}, p = e^{5}, q = e^{- 5},$ find $\frac{d r}{d s}$ .

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

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Given r=e-p2-q2,p=e5,q=e-5,finddrds.

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Engineering Physics

Rotational Dynamics

Quantum Physics

Oscillations

Work Energy and Power

Study anywhere. Anytime. Across all devices.

Given $r = e^{- p^{2} - q^{2}}, p = e^{5}, q = e^{- 5},$ find $\frac{d r}{d s}$ .