Differentiation...?

Question

Its idea is to find the gradient of a certain point on a graph, right?
How does it work?

Thanks for your time :)

Answer 1

Suppose you're given a function
y = f(x).

What the derivative helps you do is find the slope of the tangent (gradient) of a graph at a particular point.

Answer 2

An example might help:-

Example
Consider the function f(x) = x²

It might be an idea to plot this by using points:-
(-3,9) , (-2 4), (-1 ,1), (0,0), (1, 1) , (2, 4), (3,9)
The minimum turning point is (0,0) and the graph is symmetrical about the y axis.

The gradient of the graph at x is given by f `(x)
In this example, f `(x) = 2x and this gives the following gradients:-

f `(-3) = - 6 , f ` (-2) = -4 ,f `(-1) = -2
f ` (0) = 0 ,f `(1) = 2, f `(2) = 4, f `(3) = 6
This tells that the tangents to the curve change from -6 to 0 to 6

This gives the upturned U shape obtained from the point plotting.

Hopefully this example will illustrate how differentiation is used to find the gradient of tangents at different points.

Answer 3

Yes.

It works by finding the value on one side of (or the formula for the value of) and the value very near the point (or the formula there for) on the other side of the point...subtracting the values (or formulas), one from the other and dividing by the distance between the points. For example, the value I mentioned is in the y dimension and the one side or the other and the distance is in the x dimension. Therefore the calculation is the determination of a small change in y caused by a small change in x or dy/dx, or as indicated your question, the slope.

All the rules about moving exponents and subtracting one and changing from sin to cos etc are just tricks people have figured out for calculating that value when all you have is the formulas. Ya just gotta memorize them, or write them all down where you can find 'em.