Higher order derivatives are obtained by repeatedly taking the derivative of a function or its derivatives. The order of a derivative refers to how many times differentiation has been performed. To find a higher order derivative, one simply takes the derivative of the existing derivative. For example, to get the third derivative f'''(x) of a function f(x), one would take the derivative of the second derivative f''(x). Higher derivatives provide important information about the curvature and flexibility of a function at different points.

1. Higher order
derivatives

2. Basic Calculus
What you need to know already:
Basic Differentiation Rules
What you can learn here:
How to repeat the process of
differentiation to obtain
derivatives of derivatives
2

3. notice!
Do you remember the different
notations for derivatives?
3
'( )f x dy
dx
'y

4. 4
Well these are the same notations for higher power
derivatives! Any guesses on what each means?
''( )f x secthe ond derivative of f
'''y the third derivative
2
2
d y
dx
secthe ond derivative

5. 5
And to find them you just take the derivative again...and
again…if necessary!
For example to get from f’’(x) to f’’’(x) you just take the
derivative of f’’(x).
And to get from f’(x) to f(4)(x) you would just take the
derivative of f’(x) three times.

6. 1.
6
Find the second derivative of f(x) = x4 – 2x3
'( )f x  3
4x 2
6x
''( )f x  2
12x 12x[Sol]

7. 7
2. If f(x) = x 3 − 6x 5 , then
[Sol] f
(1)
(x) = 3x
2
− 30x
4
f (2)
(x) = 6x − 120x3
f
(3)
(x) = 6 − 360x
2
f (4)
(x) = −720x
f (5)
(x) = −720
f (6)
(x) = 0

8. 8
Exercise!
Please get any
paper and pencil/
ballpoint pen

9. 9

10. summary!
✘ Higher derivatives are obtained by successively
computing the derivative of a lower order
derivative.
✘ The order of a derivative refers to how many
times differentiation has been performed,
starting from the original function.
10

11. Common errors
to avoid!
✘ When looking for a pattern for the higher
derivatives of a function, don’t stop too soon:
you may need at least 5-6 derivatives before it
becomes clear.
11

12. “I have heard so many people say that you need to
forget your past and move on with life. Let me
encourage you to revisit that thought and rather
than forgetting your past, remember it and learn
from it.
YOUR PAST IS A FRIEND THAT WILL ALWAYS BE WITH
YOU TO HELP YOU IF YOU LET IT!
12

13. thanks!
13
https://www.robertosmathnotes.com/uploads/8/2
/3/9/8239617/d4-7_higher_order_derivatives.pdf