Define the disintegration rate or radioactivity of a sample and obtain the relation $R = \lambda N$ and define its different units. 

The number of disintegrating nuclei per unit time in radioactive sample is called the decay rate or radioactivity (R).

If the radioactive sample contains $\mathrm{N}$ nucleus at time $t$, then the rate of disintegration or activity $R$ is given as follows,

$\mathrm{R}=-\frac{d \mathrm{~N}}{d t}$

the negative sign indicates that as time passes activity decreases.

From law of radioactivity decay,

$-\frac{d \mathrm{~N}}{d t}=\lambda \mathrm{N}$

$\therefore \mathrm{R}=\lambda \mathrm{N}$

but $\mathrm{N}=\mathrm{N}_{0} e^{-\lambda . t}, \mathrm{R}=\lambda \mathrm{N}_{0} e^{-\lambda t}$ which is another form of the radioactive decay law and $\mathrm{m}=\mathrm{m}_{0} e^{-\lambda . t}$ is the third form.

The decay rate of a radioactive sample $\mathrm{R}$, rather than the number of radioactive nuclei is more direct experimentally measurable quantity and is given a specific name activity.

Exponential law :

$(1)$ For number $\mathrm{N}=\mathrm{N}_{0} e^{-\lambda t}$

$(2)$ For mass $\mathrm{m}=\mathrm{m}_{0} e^{-\lambda t}$ and

$(3)$ For activity $\mathrm{R}=\mathrm{R}_{0} e^{-\lambda t}$