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Derivative of Trigonometric Functions: Formulas, Rules & Easy Examples

Learn the derivatives of sine, cosine, tangent and other trig functions. Master key formulas and apply them step-by-step to solve problems quickly.

Core Theorem
Basic Trigonometric Derivatives:
ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x

ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x

ddx(tan⁡x)=sec⁡2x\frac{d}{dx}(\tan x) = \sec^2 x

ddx(cot⁡x)=−csc⁡2x\frac{d}{dx}(\cot x) = -\csc^2 x

ddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}(\sec x) = \sec x \tan x

ddx(csc⁡x)=−csc⁡xcot⁡x\frac{d}{dx}(\csc x) = -\csc x \cot x

Practice Exercises

Example 01Easy
Find ddx(sin⁡x)\frac{d}{dx}(\sin x).
NEED A HINT?
Recall the basic derivative rule for sine.
SHOW DETAILED EXPLANATION

Step 1: Identify Function

The function is sin⁡x\sin x.

Step 2: Apply Formula

Derivative of sin⁡x\sin x is cos⁡x\cos x.

Final Answer

cos⁡x\cos x
Example 02Easy
Find ddx(cos⁡x)\frac{d}{dx}(\cos x).
NEED A HINT?
Be careful with the negative sign.
SHOW DETAILED EXPLANATION

Step 1: Identify Function

The function is cos⁡x\cos x.

Step 2: Apply Formula

Derivative of cos⁡x\cos x is −sin⁡x-\sin x.

Final Answer

−sin⁡x-\sin x
Example 03Medium
Find ddx(tan⁡x)\frac{d}{dx}(\tan x).
NEED A HINT?
Tangent derivative is not cosine or sine.
SHOW DETAILED EXPLANATION

Step 1: Identify Function

The function is tan⁡x\tan x.

Step 2: Apply Formula

Derivative of tan⁡x\tan x is sec⁡2x\sec^2 x.

Final Answer

sec⁡2x\sec^2 x
Example 04Medium
Find ddx(sin⁡(3x))\frac{d}{dx}(\sin(3x)).
NEED A HINT?
Use the chain rule.
SHOW DETAILED EXPLANATION

Step 1: Outer Function

Derivative of sin⁡(u)\sin(u) is cos⁡(u)\cos(u).

Step 2: Inner Function

Derivative of 3x3x is 33.

Step 3: Multiply

cos⁡(3x)⋅3\cos(3x) \cdot 3

Final Answer

3cos⁡(3x)3\cos(3x)
Example 05Medium
Find ddx(xsin⁡x)\frac{d}{dx}(x\sin x).
NEED A HINT?
Use the product rule.
SHOW DETAILED EXPLANATION

Step 1: Product Rule

(uv)′=u′v+uv′(uv)' = u'v + uv'

Step 2: Differentiate

1⋅sin⁡x+x⋅cos⁡x1 \cdot \sin x + x \cdot \cos x

Final Answer

sin⁡x+xcos⁡x\sin x + x\cos x
Common Pitfalls
  • ⚠
    Forgetting the negative signddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x, not sin⁡x\sin x.
  • ⚠
    Confusing $\tan x$ derivativeIt is sec⁡2x\sec^2 x, NOT sec⁡x\sec x.
  • ⚠
    Ignoring chain ruleddx(sin⁡(2x))≠cos⁡(2x)\frac{d}{dx}(\sin(2x)) \neq \cos(2x) — you must multiply by 2.
  • ⚠
    Mixing up reciprocal trig functionsddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}(\sec x) = \sec x \tan x, not tan⁡x\tan x.
Gary Chang

Gary Chang

Calculus Educator

5+ Years of Calculus Teaching Experience | AP Calculus Specialist. Dedicated to helping students master calculus through step-by-step logic and clear visualizations.

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