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Derivative of Trigonometric Functions
A clear, step-by-step guide to differentiating sine, cosine, tangent, and the other trig functions, with proofs and worked examples.
What You'll Learn
The derivative of sine is cosine, and the derivative of cosine is negative sine, forming the basis for all other trig derivative rules.
Tangent, cotangent, secant, and cosecant each have their own derivative rule, all of which can be derived from the sine and cosine rules using the quotient rule.
Differentiating trig functions inside a composite expression requires the chain rule, multiplying by the derivative of the inner function.
Repeated differentiation of sine and cosine cycles through a pattern of four derivatives before repeating.
Common mistakes include mixing up the sign on the derivative of cosine and forgetting the chain rule on composite trig expressions.
What You'll Practice
1
Finding derivatives of sine and cosine functions with polynomial arguments
2
Differentiating expressions with exponents applied to trig functions vs. their arguments
3
Taking derivatives of nested trigonometric functions like sin(cos(tan(x)))
4
Using the bracket technique to simplify chain rule applications
Why This Matters
Mastering trigonometric derivatives is essential for calculus success, as these functions appear everywhere in physics, engineering, and higher mathematics. The chain rule with trig functions unlocks your ability to handle complex real-world models involving waves, oscillations, and periodic behavior.
This Unit Includes
3 Video lessons
Practice exercises
Learning resources
Skills
Trigonometric Derivatives
Chain Rule
Power Rule
Composite Functions
Bracket Technique
Co-functions
Calculus
IL Curriculum Aligned