Worksheets · Calculus

Derivatives of inverse trig functions worksheet

These six are a table to learn rather than a rule to apply, and then the chain rule does the rest. The square in the denominator is where the multiple inside ends up.

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Name

Derivatives of Inverse Trig Functions

Date Period

Differentiate each function.

  1. Differentiate inverse trig functions

    f(x)=−5tan⁡−1(5x)
  2. f(x)=−2cos⁡−1(−2x)
  3. f(x)=3tan⁡−1(−4x)
  4. f(x)=cot⁡−1(−5x)
  5. f(x)=−2cot⁡−1(−5x)
  6. f(x)=− ⁣sin⁡−1(−4x)
  7. f(x)=7cot⁡−1(3x)

  1. Differentiate inverse trig products and sec⁡−1\sec^{-1}

    f(x)=3xsin⁡−1(x)
  2. f(x)=xtan⁡−1(4x)
  3. f(x)=2x2sin⁡−1(x)
  4. f(x)=sec⁡−1(6x), x>16
  5. f(x)=xtan⁡−1(x)
  6. f(x)=sec⁡−1(7x), x>17
  7. f(x)=xtan⁡−1(2x)

  1. Differentiate compositions of inverse trig

    Find f′(x) for f(x)=csc⁡−1(4x), x>14
  2. Find f′(x) for f(x)=(tan⁡−1(2x))2
  3. Find f′(x) for f(x)=2csc⁡−1(5x), x>15
  4. Find f′(x) for f(x)=2sin⁡−1 ⁣(x3)
  5. Find f′(x) for f(x)=sin⁡−1 ⁣(x2)
  6. Find f′(x) for f(x)=(tan⁡−1(5x))3

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Derivatives of Inverse Trig FunctionsAnswer keyVersion 1

Date Period

Differentiate each function.

  1. Differentiate inverse trig functions

    f(x)=−5tan⁡−1(5x)f′(x)=−251+25x2
  2. f(x)=−2cos⁡−1(−2x)f′(x)=−41−4x2
  3. f(x)=3tan⁡−1(−4x)f′(x)=−121+16x2
  4. f(x)=cot⁡−1(−5x)f′(x)=51+25x2
  5. f(x)=−2cot⁡−1(−5x)f′(x)=−101+25x2
  6. f(x)=− ⁣sin⁡−1(−4x)f′(x)=41−16x2
  7. f(x)=7cot⁡−1(3x)f′(x)=−211+9x2

  1. Differentiate inverse trig products and sec⁡−1\sec^{-1}

    f(x)=3xsin⁡−1(x)f′(x)=3sin⁡−1(x)+3x1−x2
  2. f(x)=xtan⁡−1(4x)f′(x)=tan⁡−1(4x)+4x1+16x2
  3. f(x)=2x2sin⁡−1(x)f′(x)=4xsin⁡−1(x)+2x21−x2
  4. f(x)=sec⁡−1(6x), x>16f′(x)=1x36x2−1
  5. f(x)=xtan⁡−1(x)f′(x)=tan⁡−1(x)+x1+x2
  6. f(x)=sec⁡−1(7x), x>17f′(x)=1x49x2−1
  7. f(x)=xtan⁡−1(2x)f′(x)=tan⁡−1(2x)+2x1+4x2

  1. Differentiate compositions of inverse trig

    Find f′(x) for f(x)=csc⁡−1(4x), x>14f′(x)=−1x16x2−1
  2. Find f′(x) for f(x)=(tan⁡−1(2x))2f′(x)=4 ⁣(tan⁡−1(2x))1+4x2
  3. Find f′(x) for f(x)=2csc⁡−1(5x), x>15f′(x)=−2x25x2−1
  4. Find f′(x) for f(x)=2sin⁡−1 ⁣(x3)f′(x)=6x21−x6
  5. Find f′(x) for f(x)=sin⁡−1 ⁣(x2)f′(x)=2x1−x4
  6. Find f′(x) for f(x)=(tan⁡−1(5x))3f′(x)=15 ⁣(tan⁡−1(5x))21+25x2

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

f(x)=tan⁡−1(3x)

  1. The derivative of inverse tangent is one over one plus the inside squared.
  2. The inside is 3x, so its square is 9x squared, and its derivative is 3.
  3. So the answer is 3 over 1 plus 9x squared.

The answer is f′(x)=31+9x2.

Where students go wrong

Squaring only the x. The whole inside gets squared, so 3x becomes 9x squared, and losing the 9 is the mistake that survives all the way to the answer looking plausible.

Also called arcsin derivative, arctan derivative, inverse sine derivative or derivative of tan inverse.

What you can put on this worksheet

Derivatives of inverse trig functions
f(x)=6tan⁡−1(x) → f′(x)=61+x2
With the product rule, and inverse secant
f(x)=xtan⁡−1(3x) → f′(x)=tan⁡−1(3x)+3x1+9x2
Powers with inverse trig, and the inverse cosecant
Find f′(x) for f(x)=tan⁡−1 ⁣(4x4) → f′(x)=16x31+16x8

Questions about these worksheets

Yes. Every derivatives of inverse trig functions 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 derivatives of inverse trig functions 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 3: derivatives of inverse trig functions, with the product rule, and inverse secant and powers with inverse trig, and the inverse cosecant. Tick as many as you want and set how many of each, or let it spread them evenly.

Squaring only the x. The whole inside gets squared, so 3x becomes 9x squared, and losing the 9 is the mistake that survives all the way to the answer looking plausible.