Trigonometric derivatives describe how sine, cosine, tangent, secant, cosecant, cotangent,
and their inverse functions change. Many trig derivative problems also require the chain rule.
1. Basic trigonometric derivatives
\[
\boxed{\frac{d}{dx}\sin x=\cos x}
\]
\[
\boxed{\frac{d}{dx}\cos x=-\sin x}
\]
\[
\boxed{\frac{d}{dx}\tan x=\sec^2x}
\]
\[
\boxed{\frac{d}{dx}\sec x=\sec x\tan x}
\]
\[
\boxed{\frac{d}{dx}\csc x=-\csc x\cot x}
\]
\[
\boxed{\frac{d}{dx}\cot x=-\csc^2x}
\]
2. Chain rule with trigonometric functions
If the input is not just \(x\), use the chain rule. If:
\[
y=\sin(u)
\]
where \(u=g(x)\), then:
\[
\frac{dy}{dx}=\cos(u)\frac{du}{dx}
\]
For example:
\[
y=\sin(3x^2)
\]
\[
u=3x^2
\]
\[
u'=6x
\]
\[
\boxed{y'=6x\cos(3x^2)}
\]
3. Worked sample
Differentiate:
\[
f(x)=\sin(3x^2)+\tan x
\]
Differentiate the first term:
\[
\frac{d}{dx}\sin(3x^2)=\cos(3x^2)\cdot6x
\]
Differentiate the second term:
\[
\frac{d}{dx}\tan x=\sec^2x
\]
Combine the results:
\[
\boxed{
f'(x)=6x\cos(3x^2)+\sec^2x
}
\]
4. Inverse trigonometric derivatives
| Function |
Derivative |
Important domain note |
| \(\arcsin x\) |
\(\dfrac{1}{\sqrt{1-x^2}}\) |
\(-1<x<1\) |
| \(\arccos x\) |
\(-\dfrac{1}{\sqrt{1-x^2}}\) |
\(-1<x<1\) |
| \(\arctan x\) |
\(\dfrac{1}{1+x^2}\) |
All real \(x\) |
| \(\operatorname{arcsec}x\) |
\(\dfrac{1}{|x|\sqrt{x^2-1}}\) |
\(|x|>1\) |
| \(\operatorname{arccsc}x\) |
\(-\dfrac{1}{|x|\sqrt{x^2-1}}\) |
\(|x|>1\) |
| \(\operatorname{arccot}x\) |
\(-\dfrac{1}{1+x^2}\) |
All real \(x\) |
5. Inverse trig with chain rule
If \(u=g(x)\), then:
\[
\frac{d}{dx}\arcsin(u)=\frac{u'}{\sqrt{1-u^2}}
\]
For example:
\[
y=\arctan(5x)
\]
\[
u=5x,\qquad u'=5
\]
\[
\boxed{
y'=\frac{5}{1+25x^2}
}
\]
6. Unit circle sign reference
The unit circle helps determine signs of trig values.
| Quadrant |
\(\sin x\) |
\(\cos x\) |
\(\tan x\) |
Useful derivative sign idea |
| I |
+ |
+ |
+ |
\((\sin x)'=\cos x\) is positive. |
| II |
+ |
- |
- |
\((\cos x)'=-\sin x\) is negative. |
| III |
- |
- |
+ |
\((\tan x)'=\sec^2x\) is positive where defined. |
| IV |
- |
+ |
- |
\((\sin x)'=\cos x\) is positive. |
7. Common mistakes
- Forgetting the chain-rule multiplier, such as missing \(6x\) in \(\dfrac{d}{dx}\sin(3x^2)\).
- Writing \((\cos x)'=\sin x\) instead of \(-\sin x\).
- Writing \((\csc x)'=\csc x\cot x\) instead of \(-\csc x\cot x\).
- Forgetting that \((\tan x)'=\sec^2x\), not \(\sec x\tan x\).
- Using inverse trig formulas outside their valid domains.
- Confusing \(\sin^{-1}x\) as reciprocal sine instead of inverse sine in calculus notation.
8. Quick formula table
| Expression |
Derivative |
With chain rule |
| \(\sin u\) |
\(\cos u\) |
\(\cos(u)u'\) |
| \(\cos u\) |
\(-\sin u\) |
\(-\sin(u)u'\) |
| \(\tan u\) |
\(\sec^2u\) |
\(\sec^2(u)u'\) |
| \(\sec u\) |
\(\sec u\tan u\) |
\(\sec(u)\tan(u)u'\) |
| \(\csc u\) |
\(-\csc u\cot u\) |
\(-\csc(u)\cot(u)u'\) |
| \(\cot u\) |
\(-\csc^2u\) |
\(-\csc^2(u)u'\) |
