f(x) = sec (-x^2 - 3)

-2xsec (-x^2 - 3)tan (-x^2 - 3)

Chain rule worksheet

Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside. That last multiplication is the whole rule.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name The Chain Rule Date Period Differentiate each function. $f (x) = \sqrt{5 x^{2} + 9}$ $f (x) = \tan (3 x^{2} - 2)$ $f (x) = 5^{- 3 x^{2} - 4}$ $f (x) = \log_{3} (- x^{2} - 6)$ $f (x) = \sqrt{- 5 x^{2} + 5}$ $f (x) = \cos (2 x^{3} + 2)$ $f (x) = e^{- x^{2} + 6}$ $f (x) = \log_{10} (- 4 x^{2} + 3)$ $f (x) = \sqrt{5 x^{2} - 7}$ $f (x) = \cos (x^{3} - 8)$ $f (x) = 5^{6 x^{3} + 2}$ $f (x) = \ln (- 6 x^{3} - 4)$ $f (x) = \sqrt{- 5 x - 9}$ $f (x) = \sec (- x^{2} - 3)$ $f (x) = 5^{- 4 x^{2} + 2}$ $f (x) = \ln (6 x^{2} + 1)$ $f (x) = \sqrt{- 6 x^{2} - 7}$ $f (x) = \tan (2 x^{3} - 2)$ $f (x) = 1 0^{- 6 x^{2} - 9}$ $f (x) = \ln (2 x^{2} - 3)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Chain Rule

Date Period

Differentiate each function.

$f (x) = \sqrt{5 x^{2} + 9}$
$f (x) = \tan (3 x^{2} - 2)$
$f (x) = 5^{- 3 x^{2} - 4}$
$f (x) = \log_{3} (- x^{2} - 6)$
$f (x) = \sqrt{- 5 x^{2} + 5}$
$f (x) = \cos (2 x^{3} + 2)$
$f (x) = e^{- x^{2} + 6}$
$f (x) = \log_{10} (- 4 x^{2} + 3)$
$f (x) = \sqrt{5 x^{2} - 7}$
$f (x) = \cos (x^{3} - 8)$
$f (x) = 5^{6 x^{3} + 2}$
$f (x) = \ln (- 6 x^{3} - 4)$
$f (x) = \sqrt{- 5 x - 9}$
$f (x) = \sec (- x^{2} - 3)$
$f (x) = 5^{- 4 x^{2} + 2}$
$f (x) = \ln (6 x^{2} + 1)$
$f (x) = \sqrt{- 6 x^{2} - 7}$
$f (x) = \tan (2 x^{3} - 2)$
$f (x) = 1 0^{- 6 x^{2} - 9}$
$f (x) = \ln (2 x^{2} - 3)$



Sigma Prep · sigmaprep.io/worksheets Name The Chain RuleAnswer keyVersion 1 Date Period Differentiate each function. $f (x) = \sqrt{5 x^{2} + 9}$ $\frac{5 x}{\sqrt{5 x^{2} + 9}}$ $f (x) = \tan (3 x^{2} - 2)$ $6 x \sec^{2} (3 x^{2} - 2)$ $f (x) = 5^{- 3 x^{2} - 4}$ $- 6 x 5^{- 3 x^{2} - 4} \ln 5$ $f (x) = \log_{3} (- x^{2} - 6)$ $\frac{- 2 x}{(- x^{2} - 6) \ln 3}$ $f (x) = \sqrt{- 5 x^{2} + 5}$ $\frac{- 5 x}{\sqrt{- 5 x^{2} + 5}}$ $f (x) = \cos (2 x^{3} + 2)$ $- 6 x^{2} \sin (2 x^{3} + 2)$ $f (x) = e^{- x^{2} + 6}$ $- 2 x e^{- x^{2} + 6}$ $f (x) = \log_{10} (- 4 x^{2} + 3)$ $\frac{- 8 x}{(- 4 x^{2} + 3) \ln 10}$ $f (x) = \sqrt{5 x^{2} - 7}$ $\frac{5 x}{\sqrt{5 x^{2} - 7}}$ $f (x) = \cos (x^{3} - 8)$ $- 3 x^{2} \sin (x^{3} - 8)$ $f (x) = 5^{6 x^{3} + 2}$ $18 x^{2} 5^{6 x^{3} + 2} \ln 5$ $f (x) = \ln (- 6 x^{3} - 4)$ $\frac{- 18 x^{2}}{- 6 x^{3} - 4}$ $f (x) = \sqrt{- 5 x - 9}$ $\frac{- 5}{2 \sqrt{- 5 x - 9}}$ $f (x) = \sec (- x^{2} - 3)$ $- 2 x \sec (- x^{2} - 3) \tan (- x^{2} - 3)$ $f (x) = 5^{- 4 x^{2} + 2}$ $- 8 x 5^{- 4 x^{2} + 2} \ln 5$ $f (x) = \ln (6 x^{2} + 1)$ $\frac{12 x}{6 x^{2} + 1}$ $f (x) = \sqrt{- 6 x^{2} - 7}$ $\frac{- 6 x}{\sqrt{- 6 x^{2} - 7}}$ $f (x) = \tan (2 x^{3} - 2)$ $6 x^{2} \sec^{2} (2 x^{3} - 2)$ $f (x) = 1 0^{- 6 x^{2} - 9}$ $- 12 x 1 0^{- 6 x^{2} - 9} \ln 10$ $f (x) = \ln (2 x^{2} - 3)$ $\frac{4 x}{2 x^{2} - 3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Chain RuleAnswer keyVersion 1

Date Period

Differentiate each function.

$f (x) = \sqrt{5 x^{2} + 9}$ $\frac{5 x}{\sqrt{5 x^{2} + 9}}$
$f (x) = \tan (3 x^{2} - 2)$ $6 x \sec^{2} (3 x^{2} - 2)$
$f (x) = 5^{- 3 x^{2} - 4}$ $- 6 x 5^{- 3 x^{2} - 4} \ln 5$
$f (x) = \log_{3} (- x^{2} - 6)$ $\frac{- 2 x}{(- x^{2} - 6) \ln 3}$
$f (x) = \sqrt{- 5 x^{2} + 5}$ $\frac{- 5 x}{\sqrt{- 5 x^{2} + 5}}$
$f (x) = \cos (2 x^{3} + 2)$ $- 6 x^{2} \sin (2 x^{3} + 2)$
$f (x) = e^{- x^{2} + 6}$ $- 2 x e^{- x^{2} + 6}$
$f (x) = \log_{10} (- 4 x^{2} + 3)$ $\frac{- 8 x}{(- 4 x^{2} + 3) \ln 10}$
$f (x) = \sqrt{5 x^{2} - 7}$ $\frac{5 x}{\sqrt{5 x^{2} - 7}}$
$f (x) = \cos (x^{3} - 8)$ $- 3 x^{2} \sin (x^{3} - 8)$
$f (x) = 5^{6 x^{3} + 2}$ $18 x^{2} 5^{6 x^{3} + 2} \ln 5$
$f (x) = \ln (- 6 x^{3} - 4)$ $\frac{- 18 x^{2}}{- 6 x^{3} - 4}$
$f (x) = \sqrt{- 5 x - 9}$ $\frac{- 5}{2 \sqrt{- 5 x - 9}}$
$f (x) = \sec (- x^{2} - 3)$ $- 2 x \sec (- x^{2} - 3) \tan (- x^{2} - 3)$
$f (x) = 5^{- 4 x^{2} + 2}$ $- 8 x 5^{- 4 x^{2} + 2} \ln 5$
$f (x) = \ln (6 x^{2} + 1)$ $\frac{12 x}{6 x^{2} + 1}$
$f (x) = \sqrt{- 6 x^{2} - 7}$ $\frac{- 6 x}{\sqrt{- 6 x^{2} - 7}}$
$f (x) = \tan (2 x^{3} - 2)$ $6 x^{2} \sec^{2} (2 x^{3} - 2)$
$f (x) = 1 0^{- 6 x^{2} - 9}$ $- 12 x 1 0^{- 6 x^{2} - 9} \ln 10$
$f (x) = \ln (2 x^{2} - 3)$ $\frac{4 x}{2 x^{2} - 3}$

How to do these

$f (x) = (3 x + 1)^{4}$

The outside is something to the fourth.
That gives 4 times the parenthesis cubed.
Times the derivative of the inside, which is 3.

The answer is $12 (3 x + 1)^{3}$ .

Where students go wrong

Forgetting to multiply by the inside derivative. Without it the answer is the right shape and wrong by a factor, which is easy to miss and costs the mark.

Also called derivative of a composite function, differentiation chain rule or outside inside rule.

What you can put on this worksheet

A power or a root of a polynomial: $f (x) = (3 x + 1)^{2}$ → $6 (3 x + 1)$
A trig ratio of a function: $f (x) = \tan (3 x + 1)$ → $3 \sec^{2} (3 x + 1)$
e or another base to a function: $f (x) = e^{3 x + 1}$ → $3 e^{3 x + 1}$
A log of a function: $f (x) = \ln (3 x + 1)$ → $\frac{3}{3 x + 1}$

Questions about these worksheets

Yes. Every chain rule 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 chain rule 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.

You pick from 4: a power or a root of a polynomial, a trig ratio of a function, e or another base to a function and a log of a function. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting to multiply by the inside derivative. Without it the answer is the right shape and wrong by a factor, which is easy to miss and costs the mark.

Chain rule worksheet

Select your difficulty

The Chain Rule

The Chain RuleAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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