Second and subsequent derivatives

jeepers101

If the derivative of a function is defined as the slope of that function, what is the second and third (and so on) derivatives defined as?

Michael Collins

The second derivative is sometimes called the curvature, the third one is...the rate of change of curvature! Maybe someone else has a better name for the third derivative.

In mechanics though, if [latex] x [/latex] is a distance and [latex] t [/latex] is time, then [latex] \frac{dx}{dt} [/latex] is the velocity, [latex] \frac{d^2x}{dt^2} [/latex] is the acceleration, [latex] \frac{d^3x}{dt^3} [/latex] is the jerk, [latex] \frac{d^4x}{dt^4} [/latex] is the snap and I've heard [latex] \frac{d^5x}{dt^5} [/latex] and [latex] \frac{d^6x}{dt^6} [/latex] being refered to as crackle and pop, respectively (yes, seriously).

bnt

The second derivative is the "slope of the slope", or course, and so on. The fun starts when you apply it e.g. when it's equal to zero, that marks an inflection point (change of curvature) in the original function. The second derivative test is used to find local minima and maxima in a function. When the function has two or more variables, you can build matrices of the various derivatives, which can be useful e.g. the Hessian. I'm sure other posters can go in to much more detail if necessary.

It's also very useful in the physical sciences e.g. if f(t) is a function describing displacement (distance) w.r.t. time, then f'(t) is the velocity function, and f''(t) is the acceleration function. f'''(t) (change of acceleration w.r.t. time) is known as jerk.

edit: snap!

MathsManiac

I wouldn't be inclined (no pun intended) to say that the derivative was the slope of a function, and certainly wouldn't say that it was defined as the slope of the function. Slope is only defined (a priori, at least) for lines, so if you defined the derivative to be the slope the function, then the derivative would only be defined for linear functions, and that wouldn't be much use.

Fremen

MathsManiac wrote: »

I wouldn't be inclined (no pun intended) to say that the derivative was the slope of a function, and certainly wouldn't say that it was defined as the slope of the function. Slope is only defined (a priori, at least) for lines, so if you defined the derivative to be the slope the function, then the derivative would only be defined for linear functions, and that wouldn't be much use.

On the other hand, arguably the best way to think of the derivative is in terms of lines and slopes.

A differentiable curve has the property that as you "zoom in" on a particular point on the curve, that point and its neighbours start to look more and more like a line. The derivative at the point is the slope of that line. Here are some pictures of f(x) = x^2, zooming in on the point x=.5:









This point of view generalises very nicely to higher dimensions - a differentiable function from R^n to R^m starts to look more and more like a linear transform when you look at what it does to progressively smaller sets centered around a point p in R^n. The smaller the set, the more the function behaves like the Jacobian matrix at p.