We know that Differentiation and Integration of e^x is e^x. Doesn't it mean that the gradient and area covered under the graph of y = e^x is same. If so it is, then how?

Yeah man, the area covered by e^x from 0 to A is e^A, and that value is exactly the gradient in that point, you can prove it by solving the limit h->0 of [f(x+h)-f(x)]/h, which describes the gradient of the graph in each point

Differentiation is a method to find the gradient of the curve.

Integration is a method to find the area under the curve from (minus infinity) to x.

Yeah man, the area covered by e^x from 0 to A is e^A, and that value is exactly the gradient in that point, you can prove it by solving the limit h->0 of [f(x+h)-f(x)]/h, which describes the gradient of the graph in each point

You mean negative infinity to A