Trigonometric derivatives is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why trigonometric derivatives works
Two small-angle facts do the work: sin h divided by h heads to 1, and (cos h − 1) divided by h heads to 0. Expanding sin(x+h) with the angle addition identity and applying those two facts leaves cos x. Cosine is sine shifted over, which flips its sign, and tangent follows from the quotient rule.
What each symbol means
$x$ is the angle input.
Trigonometric derivatives: when it holds
These derivative formulas assume $x$ is measured in radians; tangent must be defined.
When it stops applying
These three statements are true for radian input only. Feed in degrees and every derivative gains a factor of π/180, so the derivative of the sine of x degrees is about 0.01745 times the cosine of x degrees, not the cosine on its own.
Trigonometric derivatives: a worked example
$\frac{d}{dx}\sin(4x)=4\cos(4x)$ by the chain rule.
The mistake to avoid
What people do: Writing the derivative of cos x as sin x with no negative sign.
Why it goes wrong: Cosine is falling on the whole interval from 0 to π, so a positive slope there contradicts the graph in front of you.
Do this instead: Remember that the three co-functions all bring a negative sign along with them when they are differentiated.
Trigonometric derivatives: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x$ is the angle input.
- Check the conditions before substituting. These derivative formulas assume $x$ is measured in radians; tangent must be defined.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where trigonometric derivatives comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about trigonometric derivatives
Why do the angles have to be in radians?
The proof leans on sin h over h approaching 1, and that is only true in radian measure. In degrees the same limit equals π/180 and every formula picks up that factor.
Is there a derivative of tan x where cosine is 0?
No. Tangent itself shoots off to infinity at those inputs, so there is no function value and no slope for the secant squared expression to report.
How do I differentiate sin of 4x?
Take the outer derivative to get cos of 4x, then multiply by the inner derivative 4, so the answer is 4 times cos of 4x.
Is there a pattern that keeps the signs straight?
Yes. Differentiating repeatedly cycles sine to cosine to negative sine to negative cosine and back, so a sign flip appears every second step.