Derivative

The first derivative is the slop of the tangent line to the function at point x. It tells if the function is increasing or decreasing.

For example:

$f(x) = 3x^3 - 6x^2 + 2x -1$

$\frac{df}{dx} = 9x^2 - 12x + 2$

$x=0$, $\frac{df}{dx}(0) = 2$, so the function is increasing at x = 0.

$x = 1$, $\frac{df}{dx}(1) = 9 - 12 + 2 = -1$, so the function is decreasing at x = 1.

The second derivative is the derivative of the derivative of the function. It tells if the function’s derivative is increasing or decreasing.

positive —> the first derivative is increasing —> the slope of the tangent line to the function is increasing as x increase. (concave up)

negative —> the first derivative is decreasing —> the slope of the tangent line to the function is decreasing as x increases. (concave down)

same example:

$f''(x) = 18x -12$

$f''(0) = 0 -12 = -12$ —> negative —> concave down at x = 0.

$f''(1) = 18 -12 = 6$ —> positive —> concave up at x = 1

If derivative is 0 or doesn’t exist, x is the critical point.

The function might be increasing/decreasing/local maximum/local minimum if the 1st derivative is 0.

However, when x is a critical point, and the 2nd derivative is:

positive —> the slope is 0, the derivative of $f(x)$ is increasing at x, concave up —> local minimum

negative —> the slope is 0, the derivative of $f(x)$ is decreasing at x, concave down —> local maximum.

0 —> the graph is changing from concave up to down or down to up, but we don’t which one.

zero-crossing detect the edge in edge detection.