# Differentiating $\nu = \dfrac{c}{\lambda}$

I am currently studying Laser Systems Engineering by Keith Kasunic. Chapter 1.2.1 Temporal Coherence says the following:

The coherence time $$\tau_c$$ over which the emitted wavelengths are considered to be in phase – that is, are temporally coherent – thus depends inversely on the absolute value of the wavelength difference $$\vert \Delta \lambda \vert$$ or frequency difference $$\Delta \nu = c \vert \Delta \lambda \vert / \lambda^2$$ [from differentiating Eq. (1.1)]: $$\tau_c \approx \dfrac{1}{2 \vert \Delta \nu \vert} = \dfrac{1}{2c} \dfrac{\lambda^2_o}{\vert \Delta \lambda \vert} \ \ \ \ \ \text{[sec]} \tag{1.3}$$ clearly showing that a narrow-spectrum laser with small $$\Delta \nu$$ has a longer time over which different frequencies propagate before they are no longer considered to be in phase. Typical numbers for a HeNe laser are used for interferometric optical testing are a coherence time $$\tau_c = 0.33$$ nsec and a coherence length $$d_c = c \tau_c = 100$$ mm for a linewidth $$\Delta \lambda = 2$$ pm (see Table 1.3). Note that the results shown in Fig. 1.11(b) – where the different wavelengths are out of phase at a time $$t \approx 10^{-8}$$ sec – are not consistent with the estimates from Eq. (1.3), as the equation assumes that $$\Delta \lambda$$ is small in comparison with the center wavelength $$\lambda_o$$.

Eq. (1.1) is given as follows:

A laser is a source of both light and heat. Light is an electromagnetic wave with a wavelength $$\lambda$$ and frequency $$\nu$$ with energy propagating at the speed of light $$c$$ in a vacuum: $$\nu = \dfrac{c}{\lambda} \ \ \ \ \ \text{[Hz]} \tag{1.1}$$

How does differentiating $$\nu = \dfrac{c}{\lambda}$$ get us (1.3)? I don't even understand how this is a function in the first place, since $$c$$ is just the speed of light in vacuum and $$\lambda$$ is the wavelength.

$$\nu=\frac{c}{\lambda}$$
$$\frac{d\nu}{d\lambda}=-\frac{c}{\lambda^2}$$ $$\frac{\Delta \nu}{\Delta \lambda}=-\frac{c}{\lambda^2}$$ $$|\Delta \nu|=\frac{c}{\lambda^2}|\Delta\lambda|$$