Differentiating Inverse Trigonometric Functions
Derive inverse-trigonometric rates on stated branches and distinguish function from derivative domains.
College Board Topic 3.4 · CED p. 73 (PDF p. 78) · CED effective Fall 2020; project course outline 2026–27.
Inverse trigonometric functions undo trig rules on chosen one-to-one branches. sin⁻¹x means arcsin x here, not 1/sin x. In radians, (arcsin x)′=1/√(1−x²) and (arccos x)′=−1/√(1−x²) for −1<x<1; (arctan x)′=1/(1+x²) for every real x. Endpoints ±1 for arcsin/arccos have no finite derivative.
Derive the sine-inverse rule rather than memorizing blindly. If y=arcsin x with −π/2≤y≤π/2, then sin y=x. Differentiating gives cos y·y′=1. On this branch cos y≥0, so cos y=√(1−x²); for |x|<1 it is positive and division gives y′=1/√(1−x²). The arccos branch [0,π] yields its negative sign; tan y=x gives sec²y·y′=1 and sec²y=1+x².
Compositions still need the chain rule. For y=arcsin(2x), y′=2/√(1−4x²) on −1/2<x<1/2. For y=arctan(x²), y′=2x/(1+x⁴). First restrict the original argument to the inverse function domain, then state where its finite derivative exists.
Other inverse-trig conventions should be stated explicitly. On arccot x∈(0,π), the derivative is −1/(1+x²). On arcsec x∈[0,π] excluding π/2, the derivative is 1/(|x|√(x²−1)) for |x|>1. On arccsc x∈[−π/2,π/2] excluding 0, it is −1/(|x|√(x²−1)) for |x|>1. These follow by differentiating cot y=x, sec y=x, or csc y=x on those branches. The absolute value records the sign dictated by the chosen branches.
Let's work through it
- Let y=arctan(3x). The inside function is 3x. Reason: this is an inverse-trig composition.
- Use the outside derivative 1/[1+(3x)²] and inside derivative 3. Reason: the inverse-tangent rule and chain rule apply.
- Obtain y′=3/(1+9x²) for every real x. Reason: the denominator stays positive and arctan has all real inputs.
Try it from memory
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Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Differentiate arccos x.
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Differentiate arccos x.
Worked answer and reason
−1/√(1−x²) for |x|<1.
The chosen arccos branch decreases with its input.
Retrieval 2
Differentiate arctan(5x).
Worked answer and reason
5/(1+25x²).
Use the inside rate 5 as well as the inverse-tangent rule.
Catch the tempting mistake
- sin⁻¹x means the reciprocal sine in every notation.
- Here it means arcsin; the reciprocal sine is csc x. State the intended meaning when notation could be ambiguous.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 3 problem 19 · No calculator · analytic
- Unit 3 problem 20 · No calculator · analytic
- Unit 3 problem 21 · No calculator · analytic
- Unit 3 problem 22 · No calculator · analytic
- Unit 3 problem 23 · No calculator · analytic
For the teacher
- Derive one inverse-trig rule using implicit differentiation and a branch sign.
- Require radians and domain checks; use the stated branches if practicing arccot, arcsec or arccsc.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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One-to-one function
A function is one-to-one when different inputs always have different outputs. Equivalently, each horizontal line meets its graph at most once. For example, x³ is one-to-one on the real numbers, whereas x² is not because 2 and −2 have the same output. This property allows a function’s inverse relation to be a function.
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Radian
A radian measures an angle by dividing the intercepted arc length by the circle’s radius. One full turn is 2π radians, so π radians equals 180 degrees. For example, an arc with length equal to its radius subtends an angle of one radian. The familiar derivative formulas for sine and cosine use radian inputs.
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Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
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Inverse function
An inverse function reverses another function’s input-output assignment. A function needs to be one-to-one on its chosen domain for an inverse function to exist. For example, x³ has inverse the cube-root function, while x² needs a domain restriction such as x≥0. The notation f⁻¹ denotes an inverse function rather than the reciprocal 1/f.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.