Derivatives of Trig Functions

Key idea

Derivatives of sin, cos, tan.

In depth

d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x. These come from the limits sin h/h → 1 and (cos h − 1)/h → 0, with x in radians.

Worked example

d/dx (sin 2x) = 2 cos 2x. d/dx (cos x) = −sin x.

Formula

Real-world uses

• Finding velocity in oscillating systems such as springs and waves
• Alternating current analysis

Common mistake

Missing the minus sign for cosine or using degrees.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Calculus

• Idea of a Limit – The value a function approaches as the input approaches a point.
• One-Sided Limits – Left and right limits must agree for the limit to exist.
• Limit Laws – Limits of sums, products and quotients are computed from the parts.
• Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
• Limits at Infinity – End behaviour of functions.
• Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.