What Is the Purpose of Differentiating a Function Twice?

[Sofie1990]

why do you differentiate twice?
In some of the questions in my homework, we have been asked to differentiate an equation twice. I understand that when you differentiate once, you are finding the gradient. When you intergrate, your finding the area. However what is the reason for differentiating twice? what exactly are you finding out?

 

[Mark44]

why do you differentiate twice?
In some of the questions in my homework, we have been asked to differentiate an equation twice. I understand that when you differentiate once, you are finding the gradient. When you inte[STRIKE]r[/STRIKE]grate, your finding the area. However what is the reason for differentiating twice? what exactly are you finding out?

You don't really differentiate an equation - you differentiate a function. As you already know, the derivative of a function gives you another function that can be used to find the slope of the tangent line at each point on the original function. Here, the derivative gives us the rate of change of the dependent variable (the function value) with respect to change in the independent variable.

If you differentiate again, you get the second derivative, which tells you the rate of change of the derivative with respect to the independent variable. That gives you information about the concavity of the original function; where it is concave up, concave down, or changing concavity.

You can keep differentiating indefinitely, but there aren't many physical properties of a graph that tie into these higher derivatives.

 

[SteveL27]

why do you differentiate twice?
In some of the questions in my homework, we have been asked to differentiate an equation twice. I understand that when you differentiate once, you are finding the gradient. When you intergrate, your finding the area. However what is the reason for differentiating twice? what exactly are you finding out?

In physics, if a function represents your position at time t, then the first derivative is the velocity and the second derivative is the acceleration.

But mathematically, if you have a function, its derivative is another function. So you can take the derivative again (assuming the 2nd derivative is defined). Given a function, it doesn't "remember" that you got it by differentiating some other function. Sort of like if you have the number 3, you can add 2 to it to get 5. Then you can add 2 again to get 7. Then again to get 9. "Adding 2" is an operation you can do to a number to get another number. Differentiating is something you can do to a function to get another function.