# Derivative of a Derivative?

#### OldManMath

##### New member
What are some real world uses of using a derivative of a derivative (in general)? I get what a derivative can tell us but I am fuzzy on what a derivative of a derivative can tell us.

Thanks

#### topsquark

##### Full Member
In Physics, for example, if we have an object that travels as a function x(t), then $$\displaystyle x'(t) = \dfrac{dx}{dt} = v(t)$$ is the speed of the object. The second derivative $$\displaystyle x''(t) = v'(t) = \dfrac{d^2x}{dt^2} = a(t)$$ is the acceleration of the object.

-Dan

#### MarkFL

##### Super Moderator
Staff member
What are some real world uses of using a derivative of a derivative (in general)? I get what a derivative can tell us but I am fuzzy on what a derivative of a derivative can tell us.

Thanks
The second derivative can tell us exactly the same kinds of things about the first derivative, that the first derivative tells us about the original function.

#### Jomo

##### Elite Member
As hinted in the previous posts, the derivative of a derivative is called the 2nd derivative.
As topsquark pointed out, the 2nd derivative of the position function is the acceleration function. Also, the 2nd derivative of the velocity function is what is known as the jerk function. This is the derivative of the acceleration function, that it it is the instantaneous change in the acceleration wrt time.
That is all physics. Just talking math, the 2nd derivative of f(x) talks about the concavity of f(x) which helps draw a pretty accurate graph of f(x).

#### OldManMath

##### New member
Thanks for the help. That clears things up for me. As a late life goal I've taken on mathematics as a hobby. As I have found out it helps me to know what something is telling me rather than calculating numbers for the sake of calculating numbers. My lesson yesterday was calculating 2nd,3rd,.... derivatives. The whole lesson I was thinking "Ok but what is this telling me?". Thanks again.

#### Subhotosh Khan

##### Super Moderator
Staff member
Thanks for the help. That clears things up for me. As a late life goal I've taken on mathematics as a hobby. As I have found out it helps me to know what something is telling me rather than calculating numbers for the sake of calculating numbers. My lesson yesterday was calculating 2nd,3rd,.... derivatives. The whole lesson I was thinking "Ok but what is this telling me?". Thanks again.
Suppose a particle is moving on a path described by y = f(x).

Then the second derivative [w.r.t. x which is f"(x)] can be used to calculate the radius of the path (think about motors running on "oval" path in a race-way).

Previous posts told you how second derivative can be used to calculate acceleration. That in conjunction with the 2nd law of motion (Newton) can be used to calculate the force needed to move the particle with that acceleration.