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2.Derivatives of Trigonometric Functions
Class 12 • Mathematics • NCERT
Derivatives of Trigonometric Functions
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This lesson covers the derivatives of standard trigonometric functions using limit definitions, as prescribed in the NCERT Class 12 syllabus.
Lesson Objectives
- Recall standard trigonometric functions.
- Learn derivatives of basic trigonometric functions.
- Apply derivative formulas in problems.
- Prepare for NCERT and board exam questions.
1. Trigonometric Functions
The six standard trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant.
sin x, cos x, tan x, cot x, sec x, cosec x
2. Derivative of sin x and cos x
Using the definition of derivative and standard limits, we obtain:
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (cos x) = −sin x
3. Derivative of tan x and cot x
The derivatives of tangent and cotangent functions are:
d/dx (tan x) = sec²x
d/dx (cot x) = −cosec²x
d/dx (cot x) = −cosec²x
4. Derivative of sec x and cosec x
The derivatives of secant and cosecant functions are:
d/dx (sec x) = sec x tan x
d/dx (cosec x) = −cosec x cot x
d/dx (cosec x) = −cosec x cot x
5. Important Observations
Trigonometric functions are differentiable wherever they are continuous in their domain.
• Derivative of sin x involves cos x
• Signs are important in derivatives
• Results are frequently used in higher applications
• Signs are important in derivatives
• Results are frequently used in higher applications
6. Important NCERT Notes
• Derivatives are obtained using standard limits
• All formulas must be memorized accurately
• Used extensively in chain rule and applications of derivatives
• All formulas must be memorized accurately
• Used extensively in chain rule and applications of derivatives
Practice Questions (NCERT)
- Find d/dx (sin x).
- Find d/dx (cos x).
- Differentiate tan x.
- Differentiate cot x.
- Find derivative of sec x.
- Find derivative of cosec x.
- Is sin x differentiable for all x?
- What is d/dx (−cos x)?
- What is d/dx (sec x + tan x)?
- Are trigonometric functions continuous?
✅ Show Answer Key
- cos x
- −sin x
- sec²x
- −cosec²x
- sec x tan x
- −cosec x cot x
- Yes
- sin x
- sec x tan x + sec²x
- Yes
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