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Solve the given problems by finding the appropriate derivatives.The radio waveguide wavelength $\lambda_{r}$, is related to its free-space wavelength $\lambda$ by$\lambda_{r}=\frac{2 a \lambda}{\sqrt{4 a^{2}-\lambda^{2}}}$where $a$ is a constant. Find $d \lambda_{r} / d \lambda$

$$\frac{8 a^{3}}{\left(4 a^{2}-x^{2}\right)^{3 / 2}}$$

Calculus 1 / AB

Chapter 23

The Derivative

Section 7

The Derivative of a Power of a Function

Derivatives

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all right here. We've been told the frequencies of different wavelengths and of different electromagnetic waves, and we want to find their difference in wavelengths. So for all electromagnetic waves, this speed of the wave, it's going to be equal to the waves wavelength times its frequency. So if we want to find the wavelength of a wave, you know that the wavelength is equal to with speed divided by the frequency. Okay, if you want to know the difference in wavelengths w one wavelength one minus wavelength to we also can call this Lambda one mind slammed the two. It's gonna be equal to see over F one Linus. See over f too. Speed of electromagnetic radiation is always the same, but the frequencies in the wavelengths could be different. Okay, this is the same thing. A C. Times one over. F one, minus one over two. Okay, so this is three. This is three times 10 to the ace meters per second times for part. A frequency one is 50 kilohertz. So this is one over 50 times 10 to the three hurts. So this isn't seconds. One over seconds is seconds. Minus one over 52 times 10 to the three hurts. Hurts is one over seconds. One over one over seconds. Two seconds. I know units get kind of crazy sometimes. So the answer to part A is one over 52 the 3 50 times, 10 to the three. So one over. Five times. 10 to the fourth, minus one over 5.2 times. 10 to the fourth times, three times 10 to be eight. So this is gonna be 200 and sir T point eight meters. Okay, that's your answer to part a week. Also around this to being about two kilometers. About 0.2 Cloutier's. All right, that's your answer to part A is 0.2 kilometers. So for part B, we're gonna use the same trick. It's gonna be C Times one over f when minus one over F two is equal to wavelength one minus. We have links to All right. See, It's still the speed of light, which is we haven't been told any different we're gonna assume is traveling through air. So its feet is three times 10 to the ace meters per second times won over 500 kilohertz. So that's 500 groups. 500 times 10 to the three. One over one over second. So this is all gonna be in units of seconds. Minus won over. 502 killers tend to the three. This is all in units of seconds. Seconds. You're gonna cancel. We're gonna be left with meters, right? This is three times 10 to me is times one over 500 times 10 to the three minus. Won over 502 times. 10 to the three. It's gonna be times point 7.968 times 10 to the minus nine. All right, so three times 10 to the eighth time, 7.9. 68% of the minus nine. It's gonna be 2.3 nine meters, which you know we can around to being around two meters. We don't feel like being very precise.

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