C12 · Unit C3 Composite, implicit and inverse derivatives
Inverse functions and inverse-trigonometric derivatives
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 3.3–3.4
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
If f is locally invertible and f minutes (a) is not equal to 0, then (f inverse) minutes (f(a)) equals 1 over f minutes (a). The input of the inverse derivative is the original output. Differentiating inverse identities gives (arcsin x) minutes equals 1/ square root of (1 minus x squared) for |x|<1 and (arctan x) minutes equals 1/(1+x squared). Arccos has the negative of the arcsin derivative.
Worked example
If f(2) equals 7 and f minutes (2) equals 3, then (f inverse) minutes (7) equals 1 over 3. For y equals arctan(2x), the chain rule gives y minutes equals 2/(1+4x squared). The denominator is positive for every real x.
Explore the mathematics
Inverse functions and inverse-trigonometric derivatives
Restrict a square function before comparing its inverse slopes.
Scroll sideways to see the full diagram.
Predict before continuing
Why must the square function have a restricted domain before inversion?
Read the complete explanation
- Why must the square function have a restricted domain before inversion?
- Reflect a point (a,a²) across y=x to obtain (a²,a).
- At a>0 the two tangent slopes are reciprocals.
- The inverse slope at x=16 is 1/(2×4)=0.125. At a=0 the reciprocal rule is not applicable.
If f(2)=7 and f′(2)=3, then (f⁻¹)′(7)=1/3. For y=arctan(2x), the chain rule gives y′=2/(1+4x²). The denominator is positive for every real x.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
Do not compute 1 over f minutes (7) in the inverse example. First recover the original input whose output is 7.
Investigate and explain
Reflect a tangent line across y equals x. Connect reciprocal slopes to exchanging horizontal and vertical changes.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.