$\lambda$ is the wavelength in, say, meters, $\nu$ is the frequency in, say, "waves per second", and c is the wave speed in meters per second. Imagine standing by a fixed point, watching the waves go by. In "T" seconds, you will have seen waves to a total length of cT meters go by. Since each wave has length $\lambda$ you will have seen $\frac{cT}{\lambda}$ waves. So $\nu T= \frac{cT}{\lambda}$ or $\nu= \frac{c}{\lambda}$.
That can also be written as $\lambda\nu= c$ or $\lambda= \frac{c}{\nu}$.
If the wave has constant wave speed (as, for example, light in vacuum) then we could easily switch between wave length and frequency using $\lambda= c\nu$: to go from
Radiance (function of wavelength)=$\frac{2ckT}{\lambda^2}$
Radiance (function of frequency)=$\frac{2kT\nu^2}{c}$
replace $\lambda^2$ with $\left(\frac{c}{\nu}\right)^2= \frac{c^2}{\nu^2}$:
$\frac{2ckT}{\lambda^2}= \frac{2ckT}{1}\frac{\nu^2}{c^2}= \frac{2ckT\nu^2}{c^2}= \frac{2kT\nu^2}{c}$.
If the wave speed, c, is NOT constant but varies with time then the relationship between wave length and frequency varies so we need that partial derivative
