Derivatives of Trigonometric Functions — Mathematics Knowledge System

Overview

The six trigonometric functions each have derivatives that can be derived from the limit definition using two key limits: lim(h→0) sin(h)/h = 1 and lim(h→0) (cos(h)-1)/h = 0. Once sin and cos are established, the derivatives of tan, cot, sec, and csc follow from the quotient rule.

Intuition

sin(x) and cos(x) are 90° out of phase, and their derivatives reflect that: differentiating sin gives cos, and differentiating cos gives -sin. The sign flip on cosine's derivative makes sense pictorially — as x increases from 0, cos(x) is decreasing, so its derivative must start negative.

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Formal Definition

Definition

The six standard trigonometric derivatives:

\frac{d}{dx}\sin(x) = \cos(x)

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

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

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

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

\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)

Derivation

Deriving d/dx sin(x) = cos(x) from the limit definition, using the angle-addition formula and the two fundamental trig limits:

\frac{d}{dx}\sin(x) = \lim_{h\to 0} \frac{\sin(x+h)-\sin(x)}{h}

Start from the definition

= \lim_{h\to 0} \frac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{h}

Angle addition formula

= \lim_{h\to 0} \left[\sin(x)\cdot\frac{\cos(h)-1}{h} + \cos(x)\cdot\frac{\sin(h)}{h}\right]

Regroup terms

= \sin(x)\cdot 0 + \cos(x)\cdot 1 = \cos(x)

Using lim(h→0)(cos h - 1)/h = 0 and lim(h→0) sin(h)/h = 1

Applications

Simple harmonic motion x(t) = A cos(ωt + φ) has velocity and acceleration expressed via sine and cosine derivatives, x'(t) = -Aω sin(ωt+φ).

Worked Examples

Apply linearity and the standard derivatives.
$f'(x) = 3\cos(x) - 5(-\sin(x)) = 3\cos(x) + 5\sin(x)$

Answer: f'(x) = 3cos(x) + 5sin(x)

Practice Problems

Difficulty 3/10

Differentiate f(x) = x²cos(x).

Difficulty 4/10

Differentiate f(x) = sec(x)tan(x).

Common Mistakes

Common Mistake

Forgetting the negative sign in d/dx cos(x) = -sin(x).

Cosine is decreasing near x=0⁺, so its derivative must be negative there — memorize the sign, it's a frequent source of error.

Common Mistake

Assuming d/dx tan(x) = sec(x) (confusing it with the derivative's relationship to secant).

The correct derivative is sec²(x), not sec(x) — verify via the quotient rule on sin(x)/cos(x).

Summary

d/dx sin(x)=cos(x), d/dx cos(x)=-sin(x) — the two building-block derivatives, derived from lim(h→0) sin(h)/h=1.
d/dx tan(x)=sec²(x), d/dx cot(x)=-csc²(x), d/dx sec(x)=sec(x)tan(x), d/dx csc(x)=-csc(x)cot(x).
All four remaining derivatives follow from sin/cos via the quotient rule.
Sign errors (especially on cosine, cotangent, and cosecant derivatives) are the most common mistake.

References

WebsiteWikipedia — Differentiation of trigonometric functions