Question

How do I visualize derivatives? I know how to derive functions, but for some reason I need a visual to fully understand the concept.

Answer 1

See answer below

Explanation:

Given: How do I visualize derivatives?

Derivatives are slope functions. When you evaluate a derivative at any location along the original function, you get the slope at that location.

If the original function is a cubic (degree = 3), then it's first derivative is a quadratic (degree = 2) function. Wherever there was a relative maximum or relative minimum in the original function, the slope will be zero which means the first derivative's `y`-value at the maximum or minimum will be equal to zero.

This would be the degree of the derivatives is you have a cubic function as the original function:

1st derivative: quadratic
2nd derivative: linear function
3rd derivative: horizontal line
4th derivative: y = 0

Here's an example:
`f(x) = x^3 + x^2 - 6x + 2` seen as pink on the graph

`f'(x) = 3x^2 + 2x - 6` seen as purple on the graph.

`f'(x)` crosses the `x`-axis when `f(x)` is both at a relative maximum and a relative minimum.

`f'(x)` has a negative `y`-value when `f(x)` is decreasing and a positive `y`-value when `f(x)` is increasing.