DerivativeCalculus.com

Trigonometric Derivatives Master

Master all trigonometric derivatives with interactive practice, memory aids, and comprehensive reference sheets

📐 Basic Trigonometric Derivatives

sin

sin(x)

d/dx [sin(x)] = cos(x)

Memory Tip:

"Sine goes to Cosine" - the simplest one!

cos

cos(x)

d/dx [cos(x)] = -sin(x)

Memory Tip:

Cosine gives negative sine - watch the minus sign!

tan

tan(x)

d/dx [tan(x)] = sec²(x)

Memory Tip:

tan' = sec² (think: "tangent gives secant squared")

cot

cot(x)

d/dx [cot(x)] = -csc²(x)

Memory Tip:

Like tan but with csc² and negative sign

sec

sec(x)

d/dx [sec(x)] = sec(x)tan(x)

Memory Tip:

sec' = sec·tan (both sec and tan appear)

csc

csc(x)

d/dx [csc(x)] = -csc(x)cot(x)

Memory Tip:

Like sec but with csc·cot and negative sign

🎯 Pattern Recognition

sin → cos

Positive transition

cos → -sin

Gains negative sign

tan → sec²

Squared function

co- → -co-

Co-functions get negative

🚀 Advanced Trigonometric Derivatives

sin⁻¹

arcsin(x)

d/dx [sin⁻¹(x)] = 1/√(1-x²)

Domain:

-1 ≤ x ≤ 1

cos⁻¹

arccos(x)

d/dx [cos⁻¹(x)] = -1/√(1-x²)

Note:

Same as arcsin but negative

tan⁻¹

arctan(x)

d/dx [tan⁻¹(x)] = 1/(1+x²)

Memory Tip:

Always positive, simple denominator

sinh

sinh(x)

d/dx [sinh(x)] = cosh(x)

Note:

No sign changes for hyperbolic!

cosh

cosh(x)

d/dx [cosh(x)] = sinh(x)

Memory Tip:

Like regular trig but no negative sign

⛓️

Chain Rule

d/dx [sin(u)] = cos(u)·u'

d/dx [cos(u)] = -sin(u)·u'

Remember:

Always multiply by derivative of inside!

🎵 Trigonometric Derivatives Song

"Sine goes to Cosine,
Cosine goes to minus Sine,
Add one to the power,
Reduce the power by one!

Tangent gives Secant squared,
Cotangent gives minus Cosecant squared,
Secant gives Secant-Tangent,
Cosecant gives minus Cosecant-Cotangent!"

✏️ Practice Problems

① Basic Practice

f(x) = 3sin(x) - 2cos(x)

Solution:
f'(x) = 3·cos(x) - 2·(-sin(x)) = 3cos(x) + 2sin(x)

② Product Rule

g(x) = x²·sin(x)

Solution:
Product rule: (f·g)' = f'·g + f·g'
f = x², f' = 2x
g = sin(x), g' = cos(x)
g'(x) = 2x·sin(x) + x²·cos(x)

③ Chain Rule

h(x) = sin(3x²)

Solution:
Chain rule: d/dx[sin(u)] = cos(u)·u'
u = 3x², u' = 6x
h'(x) = cos(3x²)·6x = 6x·cos(3x²)

④ Quotient Rule

y = tan(x)/x

Solution:
Quotient rule: (f/g)' = (f'·g - f·g')/g²
f = tan(x), f' = sec²(x)
g = x, g' = 1
y' = (sec²(x)·x - tan(x)·1)/x²

⑤ Inverse Trig

f(x) = arctan(2x)

Solution:
d/dx[arctan(u)] = u'/(1 + u²)
u = 2x, u' = 2
f'(x) = 2/(1 + (2x)²) = 2/(1 + 4x²)

⑥ Mixed Practice

y = sin(x)cos(x)

Solution:
Product rule:
y' = cos(x)·cos(x) + sin(x)·(-sin(x))
= cos²(x) - sin²(x)
Note: This equals cos(2x) by double angle formula

Your Practice Progress

Problems Attempted

Correct Answers

Accuracy Rate

Beginner

Mastery Level

⚡ Speed Quiz: 60 Seconds Challenge

🧠 Memory Aids & Mnemonics

📝

The Cycle

sin → cos → -sin → -cos → sin

4th Derivative:

4th derivative of sin(x) = sin(x)

👆

Sign Pattern

"Sine, Cosine, Negative, Negative"

Remember:

sin' = +cos, cos' = -sin

✋

Hand Trick

Thumb = sin, Index = cos

Method:

Each finger points to its derivative

🎵

Song Lyrics

"Derivative of sine is cosine,

Derivative of cosine is minus sine"

Tune:

Sing to "Twinkle Twinkle" melody

💡 Quick Reference Table

Function	Derivative	Memory Hook
sin(x)	cos(x)	"Sine goes to Cosine"
cos(x)	-sin(x)	Add negative sign
tan(x)	sec²(x)	"Tangent gives secant squared"
cot(x)	-csc²(x)	Like tan but with co- and negative