Higher order derivatives worksheet

The second derivative is the derivative of the derivative, and nothing new happens: it is the power rule again, applied to what you just wrote down.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Higher Order Derivatives Date Period Find the derivative asked for. Find the third derivative: $f (x) = 9 x^{5} - 5 x^{3} - 4 - 6 x^{- 2}$ Find the second derivative: $f (x) = - 4 x^{3} + 3 x - 3 x^{- 3}$ Find the second derivative: $f (x) = 7 x^{4} - 6 x^{2} + 6 x^{- 1}$ Find the second derivative: $f (x) = - 2 x^{3} + 5 x - x^{- 3}$ Find the second derivative: $f (x) = - 3 x^{4} + 8 x^{3} + 6 x^{- 4}$ Find the second derivative: $f (x) = - 8 x^{4} + 8 x^{3} + 2 - 5 x^{- 4}$ Find the second derivative: $f (x) = - 6 x^{3} + 5 x - 9 + 6 x^{- 2}$ Find the second derivative: $f (x) = 3 x^{4} - 2 x + 5 - 5 x^{- 3}$ Find the second derivative: $f (x) = 9 x^{3} + x - 6 x^{- 2}$ Find the third derivative: $f (x) = 8 x^{4} + 2 x + 3 - 3 x^{- 3}$ Find the third derivative: $f (x) = 3 x^{5} - 3 x - 1 - 6 x^{- 4}$ Find the second derivative: $f (x) = - 2 x^{3} + 3 x^{2} - 5 + 3 x^{- 3}$ Find the third derivative: $f (x) = - 6 x^{4} + x - 6 - 6 x^{- 3}$ Find the second derivative: $f (x) = 6 x^{4} - x^{3} - 5 x^{- 2}$ Find the second derivative: $f (x) = - x^{4} - 6 x - 9 + 3 x^{- 4}$ Find the third derivative: $f (x) = - 4 x^{5} + 5 x^{2} + x^{- 2}$ Find the third derivative: $f (x) = - 6 x^{4} - 2 x^{2} - 7 + x^{- 1}$ Find the third derivative: $f (x) = 9 x^{5} - 7 x^{4} - 5 x^{- 3}$ Find the second derivative: $f (x) = 4 x^{3} - 5 x^{2} + 6 x^{- 2}$ Find the second derivative: $f (x) = - 8 x^{4} + x + x^{- 4}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Higher Order Derivatives

Date Period

Find the derivative asked for.

Find the third derivative: $f (x) = 9 x^{5} - 5 x^{3} - 4 - 6 x^{- 2}$
Find the second derivative: $f (x) = - 4 x^{3} + 3 x - 3 x^{- 3}$
Find the second derivative: $f (x) = 7 x^{4} - 6 x^{2} + 6 x^{- 1}$
Find the second derivative: $f (x) = - 2 x^{3} + 5 x - x^{- 3}$
Find the second derivative: $f (x) = - 3 x^{4} + 8 x^{3} + 6 x^{- 4}$
Find the second derivative: $f (x) = - 8 x^{4} + 8 x^{3} + 2 - 5 x^{- 4}$
Find the second derivative: $f (x) = - 6 x^{3} + 5 x - 9 + 6 x^{- 2}$
Find the second derivative: $f (x) = 3 x^{4} - 2 x + 5 - 5 x^{- 3}$
Find the second derivative: $f (x) = 9 x^{3} + x - 6 x^{- 2}$
Find the third derivative: $f (x) = 8 x^{4} + 2 x + 3 - 3 x^{- 3}$
Find the third derivative: $f (x) = 3 x^{5} - 3 x - 1 - 6 x^{- 4}$
Find the second derivative: $f (x) = - 2 x^{3} + 3 x^{2} - 5 + 3 x^{- 3}$
Find the third derivative: $f (x) = - 6 x^{4} + x - 6 - 6 x^{- 3}$
Find the second derivative: $f (x) = 6 x^{4} - x^{3} - 5 x^{- 2}$
Find the second derivative: $f (x) = - x^{4} - 6 x - 9 + 3 x^{- 4}$
Find the third derivative: $f (x) = - 4 x^{5} + 5 x^{2} + x^{- 2}$
Find the third derivative: $f (x) = - 6 x^{4} - 2 x^{2} - 7 + x^{- 1}$
Find the third derivative: $f (x) = 9 x^{5} - 7 x^{4} - 5 x^{- 3}$
Find the second derivative: $f (x) = 4 x^{3} - 5 x^{2} + 6 x^{- 2}$
Find the second derivative: $f (x) = - 8 x^{4} + x + x^{- 4}$



Sigma Prep · sigmaprep.io/worksheets Name Higher Order DerivativesAnswer keyVersion 1 Date Period Find the derivative asked for. Find the third derivative: $f (x) = 9 x^{5} - 5 x^{3} - 4 - 6 x^{- 2}$ $540 x^{2} - 30 + 144 x^{- 5}$ Find the second derivative: $f (x) = - 4 x^{3} + 3 x - 3 x^{- 3}$ $- 24 x - 36 x^{- 5}$ Find the second derivative: $f (x) = 7 x^{4} - 6 x^{2} + 6 x^{- 1}$ $84 x^{2} - 12 + 12 x^{- 3}$ Find the second derivative: $f (x) = - 2 x^{3} + 5 x - x^{- 3}$ $- 12 x - 12 x^{- 5}$ Find the second derivative: $f (x) = - 3 x^{4} + 8 x^{3} + 6 x^{- 4}$ $- 36 x^{2} + 48 x + 120 x^{- 6}$ Find the second derivative: $f (x) = - 8 x^{4} + 8 x^{3} + 2 - 5 x^{- 4}$ $- 96 x^{2} + 48 x - 100 x^{- 6}$ Find the second derivative: $f (x) = - 6 x^{3} + 5 x - 9 + 6 x^{- 2}$ $- 36 x + 36 x^{- 4}$ Find the second derivative: $f (x) = 3 x^{4} - 2 x + 5 - 5 x^{- 3}$ $36 x^{2} - 60 x^{- 5}$ Find the second derivative: $f (x) = 9 x^{3} + x - 6 x^{- 2}$ $54 x - 36 x^{- 4}$ Find the third derivative: $f (x) = 8 x^{4} + 2 x + 3 - 3 x^{- 3}$ $192 x + 180 x^{- 6}$ Find the third derivative: $f (x) = 3 x^{5} - 3 x - 1 - 6 x^{- 4}$ $180 x^{2} + 720 x^{- 7}$ Find the second derivative: $f (x) = - 2 x^{3} + 3 x^{2} - 5 + 3 x^{- 3}$ $- 12 x + 6 + 36 x^{- 5}$ Find the third derivative: $f (x) = - 6 x^{4} + x - 6 - 6 x^{- 3}$ $- 144 x + 360 x^{- 6}$ Find the second derivative: $f (x) = 6 x^{4} - x^{3} - 5 x^{- 2}$ $72 x^{2} - 6 x - 30 x^{- 4}$ Find the second derivative: $f (x) = - x^{4} - 6 x - 9 + 3 x^{- 4}$ $- 12 x^{2} + 60 x^{- 6}$ Find the third derivative: $f (x) = - 4 x^{5} + 5 x^{2} + x^{- 2}$ $- 240 x^{2} - 24 x^{- 5}$ Find the third derivative: $f (x) = - 6 x^{4} - 2 x^{2} - 7 + x^{- 1}$ $- 144 x - 6 x^{- 4}$ Find the third derivative: $f (x) = 9 x^{5} - 7 x^{4} - 5 x^{- 3}$ $540 x^{2} - 168 x + 300 x^{- 6}$ Find the second derivative: $f (x) = 4 x^{3} - 5 x^{2} + 6 x^{- 2}$ $24 x - 10 + 36 x^{- 4}$ Find the second derivative: $f (x) = - 8 x^{4} + x + x^{- 4}$ $- 96 x^{2} + 20 x^{- 6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Higher Order DerivativesAnswer keyVersion 1

Date Period

Find the derivative asked for.

Find the third derivative: $f (x) = 9 x^{5} - 5 x^{3} - 4 - 6 x^{- 2}$ $540 x^{2} - 30 + 144 x^{- 5}$
Find the second derivative: $f (x) = - 4 x^{3} + 3 x - 3 x^{- 3}$ $- 24 x - 36 x^{- 5}$
Find the second derivative: $f (x) = 7 x^{4} - 6 x^{2} + 6 x^{- 1}$ $84 x^{2} - 12 + 12 x^{- 3}$
Find the second derivative: $f (x) = - 2 x^{3} + 5 x - x^{- 3}$ $- 12 x - 12 x^{- 5}$
Find the second derivative: $f (x) = - 3 x^{4} + 8 x^{3} + 6 x^{- 4}$ $- 36 x^{2} + 48 x + 120 x^{- 6}$
Find the second derivative: $f (x) = - 8 x^{4} + 8 x^{3} + 2 - 5 x^{- 4}$ $- 96 x^{2} + 48 x - 100 x^{- 6}$
Find the second derivative: $f (x) = - 6 x^{3} + 5 x - 9 + 6 x^{- 2}$ $- 36 x + 36 x^{- 4}$
Find the second derivative: $f (x) = 3 x^{4} - 2 x + 5 - 5 x^{- 3}$ $36 x^{2} - 60 x^{- 5}$
Find the second derivative: $f (x) = 9 x^{3} + x - 6 x^{- 2}$ $54 x - 36 x^{- 4}$
Find the third derivative: $f (x) = 8 x^{4} + 2 x + 3 - 3 x^{- 3}$ $192 x + 180 x^{- 6}$
Find the third derivative: $f (x) = 3 x^{5} - 3 x - 1 - 6 x^{- 4}$ $180 x^{2} + 720 x^{- 7}$
Find the second derivative: $f (x) = - 2 x^{3} + 3 x^{2} - 5 + 3 x^{- 3}$ $- 12 x + 6 + 36 x^{- 5}$
Find the third derivative: $f (x) = - 6 x^{4} + x - 6 - 6 x^{- 3}$ $- 144 x + 360 x^{- 6}$
Find the second derivative: $f (x) = 6 x^{4} - x^{3} - 5 x^{- 2}$ $72 x^{2} - 6 x - 30 x^{- 4}$
Find the second derivative: $f (x) = - x^{4} - 6 x - 9 + 3 x^{- 4}$ $- 12 x^{2} + 60 x^{- 6}$
Find the third derivative: $f (x) = - 4 x^{5} + 5 x^{2} + x^{- 2}$ $- 240 x^{2} - 24 x^{- 5}$
Find the third derivative: $f (x) = - 6 x^{4} - 2 x^{2} - 7 + x^{- 1}$ $- 144 x - 6 x^{- 4}$
Find the third derivative: $f (x) = 9 x^{5} - 7 x^{4} - 5 x^{- 3}$ $540 x^{2} - 168 x + 300 x^{- 6}$
Find the second derivative: $f (x) = 4 x^{3} - 5 x^{2} + 6 x^{- 2}$ $24 x - 10 + 36 x^{- 4}$
Find the second derivative: $f (x) = - 8 x^{4} + x + x^{- 4}$ $- 96 x^{2} + 20 x^{- 6}$

How to do these

$f (x) = x^{4} + 3 x^{2}$

The first derivative is 4x cubed plus 6x.
Differentiate that again.
Twelve x squared plus 6.

The answer is $12 x^{2} + 6$ .

Where students go wrong

Stopping one short, or going one too far. Count the primes in the question before you start, and count your steps as you go.

Also called second derivative, third derivative or repeated differentiation.

Questions about these worksheets

Yes. Every higher order derivatives sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the higher order derivatives questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Stopping one short, or going one too far. Count the primes in the question before you start, and count your steps as you go.

Higher order derivatives worksheet

Select your difficulty

Higher Order Derivatives

Higher Order DerivativesAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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