Formal definition of derivatives a short explanation

Note: I’m trying something new, while I’m learning new concepts, I will try to use the Feynman technique to cement my learning and share the explanations with you. Hopefully, some of these explanations can prove helpful in your learning journey.

We can turn derivatives into limits.

Here we have an example graph:

Before finding the slope of a. We can work out the slope of the general function. The plot x and x + h.

h is an arbitrary small number that can be adjusted as h approaches 0.

mistake in delta x

2 x’s cancel out in the numerator.

As h approaches zero to get the slope we can write:

Which equals f’(x)the derivative of f(x).

Now we can use the a point from earlier. We can work out the derivative of a by just plugging in the point.

Now we have made a tangent line when x=a.

Now what happens if you don’t care about slope of the function and you only care about getting one point?

You can use an alternative form below:

In visual form:

As we can see we have a tangent line between a and x.

instead of h getting smaller we have x getting smaller and getting closer to a.