/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Find \(d y / d x\). $$ x=2 e... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 4

Find \(d y / d x\). $$ x=2 e^{\theta}, y=e^{-\theta / 2} $$

Short Answer

Expert verified
The derivative of \(y\) with respect to \(x\), \(dy/dx\) is \(-e^{-\theta/2}/4e^{\theta}\) or \(-1/4e^{3\theta/2}\).

Step by step solution

01

Differentiate \(x\) with respect to \(\theta\)

Differentiate \(x\) with respect to \(\theta\), \(dx/d\theta = 2e^{\theta}\).
02

Differentiate \(y\) with respect to \(\theta\)

Differentiate \(y\) with respect to \(\theta\), \(dy/d\theta = -\frac{1}{2}e^{-\theta/2}\).
03

Application of chain rule

The chain rule states that \(dy/dx = (dy/d\theta) / (dx/d\theta)\). Plug in the values we got earlier from differentiating \(x\) and \(y\).
04

Calculate and Simplify

Calculate to find \(dy/dx = (-\frac{1}{2}e^{-\theta/2}) / (2e^{\theta})\), then simplify the equation.

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Most popular questions from this chapter

Use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ r=2 \sin (2 \cos \theta), \quad 0 \leq \theta \leq \pi $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{4}{1+2 \cos \theta}\)

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

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$$ \text { State the definition of a smooth curve } $$
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