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Calculus⏱ 6 min read

Trigonometric Derivatives

d/dx(sin x) = cos x

📖 All Six Trig Derivatives

These are the derivatives of all six basic trigonometric functions. Memorising these is essential for calculus.

Function Derivative
sin x cos x
cos x −sin x
tan x sec²x
cot x −csc²x
sec x sec x · tan x
csc x −csc x · cot x

🔤 Key Identities to Remember

sec x
Secant = 1/cos xReciprocal of cosine
csc x
Cosecant = 1/sin xReciprocal of sine
cot x
Cotangent = cos x / sin xReciprocal of tangent

📝 Example — Differentiating with Trig

Differentiate: y = 3sin x − 2cos x + tan x

1
Differentiate term by termd/dx(3sin x) = 3cos x
2
Derivative of -2cos xd/dx(-2cos x) = -2·(-sin x) = 2sin x
3
Derivative of tan xd/dx(tan x) = sec²x
✓
Answer: dy/dx = 3cos x + 2sin x + sec²x

🧮 Interactive Calculator

Enter an angle in radians to compute derivative values at that point.

💡 Memory Pattern

sin ↔ cos:

sin and cos are each other's derivatives (with a sign change for cos→sin direction)

Negative signs:

cos, cot, and csc all have negative derivatives. The "co-" functions are negative.

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🔗 Related Formulas

🎯 Quick Fact

The derivative of sin x = cos x can be proven from first principles using the limit definition of the derivative and the squeeze theorem.