The Four Derivatives You Need to Know
You already know the basic derivatives of sine and cosine. The remaining four trigonometric functions have derivative rules that are just as important: tangent, cotangent, secant, and cosecant.
The key is not to memorize these as four completely unrelated formulas. They come in natural pairs, and their signs follow a pattern that makes them easier to remember.
Tangent
Positive derivative; the derivative is secant squared.
Cotangent
Negative derivative; the derivative is cosecant squared.
Secant
The function reproduces itself and picks up a tangent factor.
Cosecant
The function reproduces itself with a negative sign and cotangent factor.
A Pattern Worth Memorizing
There are two especially useful pairs:
- Tangent and cotangent: their derivatives are squares of their reciprocal functions.
- Secant and cosecant: their derivatives contain the original function multiplied by its corresponding cofunction.
Notice the signs: tan is positive, cot is negative, sec is positive, and csc is negative. Remembering that alternating sign pattern prevents one of the most common errors on these problems.
Why Is the Derivative of Tangent Secant Squared?
You can derive the tangent rule instead of treating it as a mysterious formula. Start by writing tangent as a quotient:
Now apply the quotient rule:
Use the Pythagorean identity :
So:
This gives the familiar result:
Why Is the Derivative of Cotangent Negative?
Cotangent can be written as cosine divided by sine:
Applying the quotient rule gives:
Using , the result simplifies to:
The negative sign is essential. A surprisingly common mistake is to remember instead of the correct formula.
Why Do Secant and Cosecant Keep the Original Function?
The secant derivative follows directly from the reciprocal relationship:
Using the quotient rule,
and simplifying with the identities and gives:
Similarly, because , differentiation gives:
So the two rules are:
Using the Chain Rule
The formulas above are for functions of . But AP Calculus problems often put another function inside the trigonometric function. When that happens, you need the chain rule.
For example, if , the derivative is not merely . You must multiply by the derivative of the inside:
The same idea works for all four functions:
Worked Examples
Differentiate .
Differentiate .
Differentiate .
Differentiate .
Differentiate .
Evaluating Trigonometric Derivatives at a Point
AP Calculus problems frequently ask you to find a derivative value such as . In these problems, the most important step is often recognizing which trigonometric values you need.
Let . Find .
Common Mistakes
The derivatives of cotangent and cosecant are negative. The safest memory pattern is tan positive, cot negative, sec positive, csc negative.
Secant differentiates to , while cosecant differentiates to . The original function stays in both answers, but the companion function and sign change.
For , the derivative is not just . You must multiply by the derivative of the inside function.
The derivative of tangent is , not . Likewise, the derivative of cotangent is , not .
A Fast AP Calculus Memory Table
A useful verbal shortcut is:
- Tangent → secant squared
- Cotangent → negative cosecant squared
- Secant → secant tangent
- Cosecant → negative cosecant cotangent
Practice Problems
Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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