Lesson 3: Derivatives of Trig Functions
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State the derivatives of sin x and cos x from the limit definition
- State and use the derivatives of all six trig functions
- Combine trig derivatives with the Chain Rule, Product Rule, and Quotient Rule
Derivatives of Sine and Cosine
Using the limit definition of the derivative along with the special limits limh→0 (sin h)/h = 1 and limh→0 (cos h − 1)/h = 0, we can prove:
The Two Fundamental Trig Derivatives
d/dx [sin x] = cos x
d/dx [cos x] = −sin x
Notice the minus sign on cosine. A helpful mnemonic: derivatives of functions that start with "co-" (cosine, cotangent, cosecant) all pick up a negative sign.
Example 1: Basic Trig Derivatives
Find d/dx [3 sin x − 2 cos x].
Solution: 3 cos x − 2(−sin x) = 3 cos x + 2 sin x
Derivatives of the Other Four Trig Functions
Using the Quotient Rule on the definitions tan x = sin x / cos x, etc., we derive:
All Six Trig Derivatives
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| cot x | −csc² x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
Example 2: Deriving d/dx [sec x]
Write sec x = 1/cos x and apply the Quotient Rule (or think of it as (cos x)−1 and use the Chain Rule):
d/dx [1/cos x] = −(−sin x)/cos² x = sin x / cos² x = (1/cos x)(sin x/cos x) = sec x tan x
Combining with the Chain Rule
Example 3: Chain Rule with Trig
Find d/dx [tan(5x)].
Outer: tan u. Inner: u = 5x.
d/dx [tan(5x)] = sec²(5x) · 5 = 5 sec²(5x)
Example 4: Product Rule + Trig
Find d/dx [x² cos x].
Product Rule: 2x cos x + x²(−sin x) = 2x cos x − x² sin x
Example 5: Nested Trig Composition
Find d/dx [sec²(3x)].
Rewrite: [sec(3x)]². Outer: u². Inner: sec(3x).
General Power Rule: 2 sec(3x) · d/dx[sec(3x)]
Chain Rule for sec(3x): sec(3x) tan(3x) · 3
Combine: 2 sec(3x) · 3 sec(3x) tan(3x) = 6 sec²(3x) tan(3x)
Key Takeaways
- Memorize the six trig derivatives. The "co-" functions all have a negative sign.
- The pairs are: sin/cos, tan/sec, cot/csc.
- When trig functions have a composite argument (like sin(3x)), always apply the Chain Rule.
- Most trig differentiation problems combine trig derivatives with the Product, Quotient, or Chain rules.
Check Your Understanding
1. Find d/dx [sin x cos x].
2. Find d/dx [csc(4x)].
3. Find d/dx [tan x / x].
4. Find d/dx [cos³(2x)].
Ready for More?
Next Lesson
Lesson 4 covers implicit differentiation for curves defined by equations like x² + y² = 25.
Start Lesson 4