Question

Can anybody explain the function composition of the definition of a derivative in Laymen terms.

Answer 1

Do you mean the definition of the derivative the composition of functions, like f(g(x))?

Answer 2

or [f(x+h)-f(x)]/h?

Answer 3

So the point of this expression is really quite simple. It is literally the same expression as the one for slope of a line.
\[\frac{ f(x+h)-f(x) }{ (x+h)-(x) }\]
The "m" if you will of "y=mx+b". The only difference is we are now plugging in the function (which is just a y value) for the rise and using the same old x for the run. Rise over run. We take the limit as h goes to zero because on most graphs, like y=xsquared the slope changes wherever you are.



Since the derivative and original function both use the same "x" values, that means the y value of x on the derivative is the slope of the function on that same x value of the original graph.

So, at the bottom of xsquared the graph has no slope, it has a horizonal line there for a slope, the rise over run is 0, right? Similarly, at x=1 the slope is 2 as seen from the graph of the derivative.

Hopefully that helps. It will take a little thinking and experimentin.