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

Graphical or geometric meaning

At an angle where sine has a local maximum, cosine is zero, matching the horizontal tangent. Cosine begins at a maximum and initially decreases, which explains the negative sign in $(\cos x)'=-\sin x$.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate an equivalent identity and compare results.
• Evaluate at special angles to see whether the sign and zeros make sense.
• Use a short numerical difference quotient in radian mode.

Use structure instead of memorizing variants

Because $\tan x=\sin x/\cos x$, the quotient rule recovers $\sec^2x$ and shows why tangent is differentiable only where $\cos x\ne0$. These rules assume radians; degree measure introduces a conversion factor. With a composite angle, retain the inner derivative, as in $(\sin u)'=\cos(u)u'$. A quick sign check comes from the unit circle: near zero, sine increases and cosine has a horizontal tangent, matching $\cos0=1$ and $-\sin0=0$. Simplify identities only when doing so reduces product or quotient complexity, and keep excluded points visible even if an algebraic factor cancels.

Practice

1. Differentiate $\cos(3x)$.
2. Differentiate $\tan(x^2)$.
3. Differentiate $\sec x$.

Connections and next steps

• Chain Rule
• Basic Differentiation Rules
• Implicit Differentiation