Essential Knowledge 2.1D2

Essential Knowledge 2.1D2 Students will know that higher order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include

, f ''(x), and y''. Higher order derivatives can be denoted or f ⁽ⁿ⁾(x).

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In Essential Knowledge 2.1D1, we found the acceleration of a falling ball. Our equations were as follows.

y = -16t² + 1000

y' = v = -32t

v' = a = -32

We said acceleration was a higher order derivative because it was a derivative of a derivative

Now we will learn some other symbols. The three styles for higher order derivative symbols, just like for first order derivatives, are equation prime notation, function prime notation, and Leibniz's notation

Equation prime notation

We started with

y = -16t² + 1000

and found that

y' = -32t

Instead of talking about velocity, we could just take the derivative of this equation again as it is. You would show this like so

y'' = -32 "y double prime" = -32

"The second derivative of y" = -32

We could take another derivative if we wanted to, and it would look like this

y''' = 0 "y triple prime" = 0

"The third derivative of y" = 0

Function prime notation

We could have started with

f(t) = -16t² + 1000

and found that

f'(t) = -32t

Instead of talking about velocity, we could just take the derivative of this function again as it is. You would show this like so

f''(t) = -32 "f double prime of t" = -32

"The second derivative of f" = -32

We could take another derivative if we wanted to, and it would look like this

f'''(t) = 0 "f triple prime of t" = 0

"The third derivative of f" = 0

Leibniz's Notation

We could have started with

y = -16t² + 1000

and found that

Instead of talking about velocity, we could just take the derivative of this equation again as it is. You would show this like so

" " = -32

"The second derivative of y with respect to t" = -32

We could take another derivative if we wanted to, and it would look like this

" " = 0

"The third derivative of y with respect to t" = 0