Differentiation gave mathematics a language for changing slopes. Integration gave it a language for accumulating quantities and areas. Together, they transformed the study of motion, change, and modern analysis.

Differentiation original formula f(x+h)-f(x)/h. Where f(x) is value of x at (0,0) for cordinate system (basically f(0)) and h is the horizontal distance. d/dx(x^2) at x=1,f'(x^2)= f(0+1)-f(0)/1 = 1, at x=2 ,f'(x^2)= f(0+2)-f(0)/2= 4-0/2=2. So d/dx(x^2) =x

f'(x) =f(h)-f(0)/h , h is horizontal distance which is equal to x. f'(x) = (f(x) -f(0))/ x

and the fundamental theorem says they undo each other. take the slope of the area under the curve up to x and you get the curve back