Conditions before computation
The formulas used here assume radian inputs. For arcsin(u), the ordinary finite derivative formula requires |u|<1; the square-root denominator can vanish at the endpoints. For arctan(u), the denominator 1+u² is positive for every real u. Multiplying by u′ is a separate chain-rule step in every case. If an answer asks for a coefficient of √3, keep that convention separate from the full derivative value.
Worked example with a method check
f(x)=tan(3x). Find f′(π/12). The chain rule gives f′(x)=3sec²(3x). At the stated x, the angle is π/4 and sec²(π/4)=2. The derivative is 6.
Compare two methods
For f(x)=tan(3x), f′=3sec²(3x). For g(x)=arctan(3x), g′=3/[1+9x²]. These functions do not share a derivative formula merely because both names contain tan. At x=0 both derivatives happen to equal 3, so testing only that point would not distinguish the rules. A correct method should also explain behavior at other inputs and identify excluded points.
Transfer: solve before reading
Find the derivative of arcsin(x/5) at zero and the derivative of arctan(x/5) at x=5. Explain why the chain factor is the same but the final answers differ.
Transfer reasoning and answer
Both inner derivatives are 1/5. At zero the arcsine denominator is √(1−0)=1, giving 1/5. At x=5 the arctangent denominator is 1+1²=2, giving 1/10. The difference comes from the outer derivative and its evaluated input.
Practice the next representation
Work all four task families linked to this lesson. For the illustrated items, state which numerical labels and relationships you used. For a written extension, compare each line of your solution with its original self-assessment criteria; these criteria are practice feedback, not an official AP score.
f(x)=tan(3x). Find f′(π/12).
- The chain rule gives f′(x)=3sec²(3x).
- At the stated x, the angle is π/4 and sec²(π/4)=2.
- The derivative is 6.
Compare the other answer choices
Omit the inner derivative.
Apply the inner derivative twice.
Use the sign rule for a cosine derivative.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
(a) State the relationship or theorem that makes this calculation valid, including its relevant conditions. (b) Show the intermediate computation and give the requested result. (c) Explain why one plausible competing method would fail for this problem.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB and BC Course and Exam DescriptionCalculus Volume 1: Fundamental Theorem of Calculus