Question

Are we following darwin concept in evolution?

Answer 1

Yes
Darwin's theory has two aspects to it, namely Natural Selection and Adaptation, that work together to shape the inheritance of alleles (forms of a gene) within a given population. Darwin made the following five fundamental observations, from which three inferences can be drawn.

Darwin's observations

All species have such great potential fertility that their population size would increase exponentially if all individuals that are born go on to reproduce successfully.
Populations tend to remain stable in size, except for seasonal fluctuations.
Environmental resources for things such as food and shelter are limited.
Individuals of a population vary extensively in their characteristics (to the extent that no two individuals are exactly alike) which impacts upon their own ability to survive and reproduce.
Much of this variation is genetic and is therefore heritable.
Inferences drawn from observations

Due to the limited resources, there is a struggle for existence among individuals - often with only a fraction of offspring surviving through each generation to reproduce successfully.
It is not a random process that determines which individuals will reproduce and which will not, as it depends in part on the genetic / hereditary constitution of those surviving individuals. Those individuals whose inherited characteristics best suit them to that environment in which they live are likely to have more offspring than those that are not so well adapted to the environment. This, by definition, is natural selection.
The unequal ability between individuals to survive and reproduce will lead to gradual evolution of the population, with favourable characteristics accumulating over the generations through natural selection.