The relation between the derivative of the function and the integral of the inverse function to the derivative function

In the first what is the calculus? Before defining the calculus we can divide it into two categories; the first is differentiation, but the second is the integration. The differentiation: that category relates with the rate of change and the slope of the graphs, whereas we use the limits in finding the derivative of any curve: -The integration: that category relates with the area under and between the graphs, whereas appeared many theorems like Riemann integration, Monte Carlo integration and the fundamental theorem of calculus. -The differentiation is connected with integration by The fundamental theorem of calculus that tells us that: -The differential calculus and integral calculus is well-connected by many thing and from the new things that the derivative of a function can be found by knowing the integral of the inverse function to the derivative function, and that can be understood from the following equation: -The previous equation told us that: 1-we can find the derivative of any curve without taking the limits. 2-That equation is an good evidence for the strong connection between the differentiation and integration.