Trig Derivative Rules- Comprehensive Reference
The Trig Derivative Rules You Actually Need
Calculus without trig derivatives is like cooking without knowing how to chop onions. You can do it, but everything takes forever. This is your complete reference for every trig derivative rule you'll encounter in calculus.
No philosophical rambling. Just the rules, why they work, and how to use them.
Basic Trig Derivatives
Every trig function has a derivative. Memorize these six. They form the foundation for everything else.
- 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)
That's it. Six rules. The rest is just applying other calculus concepts on top of these.
Why These Derivatives Work
The derivative of sin(x) is cos(x) because of the limit definition of the derivative and the angle addition formulas. When you work through the math:
d/dx[sin(x)] = lim(h→0) [sin(x+h) - sin(x)]/h
Using sin(x+h) = sin(x)cos(h) + cos(x)sin(h), the limit simplifies to cos(x).
Same process gives you cos(x) = -sin(x). The other four follow from the quotient rule and reciprocal identities.
You don't need to derive these on exams. You need to apply them.
Derivatives of Inverse Trig Functions
Inverse trig functions have their own derivatives. These come up less often but show up in integration problems constantly.
- arcsin(x) → 1/√(1-x²)
- arccos(x) → -1/√(1-x²)
- arctan(x) → 1/(1+x²)
- arccot(x) → -1/(1+x²)
- arcsec(x) → 1/(|x|√(x²-1))
- arccsc(x) → -1/(|x|√(x²-1))
The absolute value in arcsec and arccsc exists because these functions have restricted domains. Most textbooks ignore it in simplified form, but it's technically there.
The Chain Rule: Your New Best Friend
Most trig derivatives in real problems aren't simple sin(x). They're sin(3x), tan(x²), or sec(5x+1). That's where the chain rule comes in.
Chain Rule Formula
If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x)
In trig terms: the derivative of sin(u) where u is a function of x is cos(u) · du/dx.
Examples
sin(3x)
u = 3x, du/dx = 3
Derivative = cos(3x) · 3 = 3cos(3x)
tan(x²)
u = x², du/dx = 2x
Derivative = sec²(x²) · 2x = 2x·sec²(x²)
cos(5x³ + 2x)
u = 5x³ + 2x, du/dx = 15x² + 2
Derivative = -sin(5x³ + 2x) · (15x² + 2)
Product and Quotient Rule Applications
When trig functions multiply or divide, use the product and quotient rules.
Product Rule
d/dx[f·g] = f'·g + f·g'
Example: x²·sin(x)
f = x², f' = 2x
g = sin(x), g' = cos(x)
Derivative = 2x·sin(x) + x²·cos(x)
Quotient Rule
d/dx[f/g] = (f'·g - f·g')/g²
Example: sin(x)/x
f = sin(x), f' = cos(x)
g = x, g' = 1
Derivative = [cos(x)·x - sin(x)·1]/x² = (x·cos(x) - sin(x))/x²
Quick Reference Table
| Function | Derivative | Notes |
|---|---|---|
| sin(x) | cos(x) | Base case |
| cos(x) | -sin(x) | Negative sign |
| tan(x) | sec²(x) | Never negative |
| cot(x) | -csc²(x) | Always negative |
| sec(x) | sec(x)tan(x) | Product of two |
| csc(x) | -csc(x)cot(x) | Negative product |
| arcsin(x) | 1/√(1-x²) | Domain: -1 < x < 1 |
| arccos(x) | -1/√(1-x²) | Negative of arcsin |
| arctan(x) | 1/(1+x²) | No domain restrictions |
How To: Taking Trig Derivatives in Practice
Here's the step-by-step process for any trig derivative problem.
Step 1: Identify the Structure
Is it a single trig function? A composite? A product? A quotient?
- sin(x), cos(2x) → basic or chain rule
- sin(x)·cos(x) → product rule
- sin(x)/x → quotient rule
Step 2: Apply the Right Rule
For sin(4x² + 1):
Inner function: u = 4x² + 1
Outer derivative: cos(u)
Chain multiplier: du/dx = 8x
Final answer: 8x·cos(4x² + 1)
Step 3: Simplify
Factor common terms. Move coefficients to the front. Don't leave unnecessary parentheses.
Step 4: Check for Patterns
sin²(x) derivative uses chain rule twice: 2·sin(x)·cos(x) = sin(2x)
tan²(x) derivative: 2·tan(x)·sec²(x)
Common Mistakes That Cost Points
- Forgetting the chain rule — sin(3x) is not cos(3x), it's 3cos(3x)
- Dropping the negative on cos(x) — the derivative is -sin(x), not sin(x)
- Confusing cot and csc — cot derivative is -csc²(x), csc derivative is -csc(x)cot(x)
- Wrong quotient rule setup — remember f'g - fg', not fg' - f'g
- Ignoring domain restrictions — arcsec and arccsc need |x| in the denominator
Higher-Order Trig Derivatives
Sometimes you need the second, third, or fourth derivative. These cycle through a pattern.
sin(x): cos(x) → -sin(x) → -cos(x) → sin(x) → repeats every 4
cos(x): -sin(x) → -cos(x) → sin(x) → cos(x) → repeats every 4
For higher derivatives of sin(ax) or cos(ax), multiply by a each cycle and account for the negative signs.
The Bottom Line
Six basic trig derivatives. Four inverse trig derivatives. Chain rule for compositions. Product and quotient rules for combinations. That's the entire list.
Practice applying these rules until you don't have to think about which one to use. The pattern recognition comes with reps, not with reading articles.