Derivatives of Trigonometric Functions

The central idea

With angles measured in radians, $(\sin x)'=\cos x$, $(\cos x)'=-\sin x$, $(\tan x)'=\sec^2x$, $(\cot x)'=-\csc^2x$, $(\sec x)'=\sec x\tan x$, and $(\csc x)'=-\csc x\cot x$. For $g(u(x))$, multiply by $u'$ through the chain rule.

Definitions, hypotheses, and notation

The radian convention is mathematical, not cosmetic. In degrees, the small-angle ratio $\sin h/h$ tends to $\pi/180$, so extra conversion constants appear in every derivative. Radians normalize arc length so that this limit is one, producing the clean standard table. Any numerical derivative check must therefore place the calculator in radian mode.

The six formulas can be reduced to fewer facts. Tangent, cotangent, secant, and cosecant are quotients or reciprocals of sine and cosine, so their derivatives follow from quotient and reciprocal rules plus identities. Deriving a forgotten formula is safer than guessing its sign. Domain exclusions, such as $\cos x=0$ for tangent, remain in force after differentiation.

Conceptual meaning

Sine and cosine derivatives reflect quarter-cycle phase shifts. On the unit circle, the velocity vector of $(\cos x,\sin x)$ is $(-\sin x,\cos x)$, tangent to the circle and perpendicular to the radius.

A dependable method and decision rule

1. Convert degree-based input to radians before using the standard formulas.
2. Identify the outer trigonometric function and its full argument.
3. Apply the correct trig derivative, including its sign.
4. Multiply by the derivative of the argument.
5. Combine with product, quotient, or chain rules and simplify using identities only as helpful.

Fully worked example

Common mistakes and why they fail