Trig Derivatives: Step-by-Step Guide
The six trig functions have memorable derivatives: d/dx sin x = cos x, d/dx cos x = -sin x, d/dx tan x = sec^2 x, d/dx cot x = -csc^2 x, d/dx sec x = sec x tan x, d/dx csc x = -csc x cot x. With composite arguments, apply the chain rule.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Identify the trig function and its argument. Determine which of the six trig functions appears and what is inside.\[f(x) = \sin(g(x))\]
- Recall the basic derivative. Use the corresponding rule from the table.\[\frac{d}{dx}\sin x = \cos x,\;\frac{d}{dx}\tan x = \sec^2 x\]
- Apply the chain rule. Multiply by the derivative of the inner function g'(x).\[\frac{d}{dx}\sin(g(x)) = \cos(g(x)) \cdot g'(x)\]
- Simplify. Combine factors and apply identities if it produces a cleaner form.\[\sin(2x)\cdot 2 = 2\sin(2x)\]
Worked example
Find d/dx [sin(3x^2 + 1)].
Step 1: Outer function is sin; inner function g(x) = 3x^2 + 1.
Step 2: d/dx sin(u) = cos(u).
Step 3: g'(x) = 6x. By chain rule: d/dx sin(3x^2 + 1) = cos(3x^2 + 1) * 6x.
Step 4: Final: 6x * cos(3x^2 + 1).
Verification: Numerical check at x = 1: derivative ~= 6 * cos(4) ~= 6 * (-0.6536) ~= -3.92. Finite-difference (sin(4.0001) - sin(4))/0.0001 also ~= -3.92.
Tips and common mistakes
- Memorize the six derivatives - three negatives are paired with co-functions (cos, cot, csc).
- Always apply the chain rule when the argument is anything other than x.
- d/dx tan x = sec^2 x is critical for trig substitution integrals.
- Watch the sign on cos and csc derivatives - they include a leading minus.
- For products like x * sin x, use the product rule then a derivative rule.
- For sin^n(x), treat as (sin x)^n and use the chain rule.
- Higher derivatives of sin/cos cycle every 4 steps: sin -> cos -> -sin -> -cos -> sin.
- Double-check by plugging a value and using a finite-difference approximation.