Why Inverse Trigonometric Derivatives Matter
You've already learned the derivatives of the basic trigonometric functions. But AP Calculus also expects you to differentiate their inverse functions: arcsine, arccosine, and arctangent.
The notation can look intimidating at first. For example, does not mean . Instead, it means the inverse function of sine: the angle whose sine is .
The good news is that there are only three formulas you really need to know for AP Calculus:
Once you know these formulas, the main challenge is recognizing when to use them and remembering to apply the chain rule when the input is something more complicated than just .
Where the Formulas Come From
You don't have to memorize these formulas without understanding them. They come directly from the relationship between a function and its inverse.
Start with:
Because and are inverse functions, we can rewrite this as:
Now differentiate both sides with respect to . The left side requires the chain rule:
Solving for the derivative of the inverse function gives:
For sine, cosine, and tangent, the corresponding identities can be simplified using the Pythagorean identity. That's where the familiar inverse-trigonometric derivative formulas come from.
The Three Essential Formulas
1. Derivative of Arcsine
If , then:
The square root is important. The expression underneath it is , not .
2. Derivative of Arccosine
If , then:
Notice the negative sign. This is one of the most common places students make an error.
3. Derivative of Arctangent
If , then:
Unlike arcsine and arccosine, the arctangent formula has no square root.
The Chain Rule Connection
The formulas above assume that the input is simply . But AP Calculus frequently gives you an inverse trigonometric function with a more complicated input.
For example:
Here, the input to the inverse sine is , not just . So after applying the inverse-sine derivative formula, you must multiply by the derivative of the inside function.
This is simply the chain rule applied to an inverse trigonometric function.
Worked Examples
Differentiate .
Differentiate .
Differentiate .
Differentiate .
Differentiate .
Understanding the Difference Between Inverse and Reciprocal
The notation means the inverse function of sine, not . Likewise, means inverse tangent, not the reciprocal of tangent.
The reciprocal of sine is cosecant:
The inverse sine function is:
These are completely different functions with completely different derivatives.
Common Mistakes
The derivative of is negative. A quick memory aid is that arcsine is positive while arccosine is negative.
For , the answer is not just the basic inverse-sine formula. You must multiply by , the derivative of the inside function.
For arcsine and arccosine, the denominator contains . If the input is , substitute the entire expression into the formula.
Remember: is an inverse function, while is a reciprocal function.
A Reliable AP Calculus Strategy
- Identify the inverse trig function. Is it arcsine, arccosine, or arctangent?
- Write the corresponding formula. Don't try to do everything mentally.
- Replace the input with the entire inside expression.
- Multiply by the derivative of the inside expression. This is the chain rule.
- Simplify only after the derivative is correct.
Practice Problems
Differentiate .
Show answer
Differentiate .
Show answer
Differentiate .
Show answer
Differentiate .
Show answer
Differentiate .
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon