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Differentiation of Trigonometric Functions — Formulas & Rules

#Trigonometry

TL;DR

The differentiation of trigonometric functions gives the six core rules:

1. ( \frac{d}{dx}\sin x = \cos x )
2. ( \frac{d}{dx}\cos x = -\sin x )
3. ( \frac{d}{dx}\tan x = \sec^2 x )
4. ( \frac{d}{dx}\cot x = -\csc^2 x )
5. ( \frac{d}{dx}\sec x = \sec x \tan x )
6. ( \frac{d}{dx}\csc x = -\csc x \cot x )

All six follow from the sine and cosine derivatives via the quotient rule. This article proves them from first principles and shows where students slip.

A Swing With No Ending

Galileo timed a chandelier swinging during mass in 1583 and noticed something odd — the swing took the same time whether the arc was wide or narrow. That observation became calculus's first physical example of position → velocity → acceleration under sine and cosine. The reason the derivative of ( , \sin x ) is ( \cos x ) isn't a notation trick. It's that velocity always sits one quarter-period ahead of position in any oscillating system.

What Is Differentiation of Trigonometric Functions?

Differentiation of trigonometric functions is the process of finding the derivative — the instantaneous rate of change — of ( , \sin x , ), ( , \cos x , ), ( , \tan x , ), and their reciprocal partners. The derivatives come in clean pairs: each function's derivative is closely related to another trig function (or its negative). All six rules are proved from two foundational limits and the quotient rule.

The Six Derivative Rules

Pattern to lock in. Every "co-" function (cosine, cotangent, cosecant) has a negative sign in its derivative. The three "non-co" functions don't. That single rule recovers half the table on exam day if memory fails.

These rules assume ( x ) is measured in radians. If ( x ) is in degrees, every derivative picks up a factor of ( \frac{\pi}{180} ) — which is why no calculus textbook works in degrees. We come back to this in the mistakes section.

Proof From First Principles — Derivative of ( \sin x )

Starting from the limit definition:

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

Apply the sum identity ( \sin(x+h) = \sin x \cos h + \cos x \sin h ):

[ = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h} ]

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

Two foundational limits do the rest:

• ( \lim_{h \to 0}\frac{\sin h}{h} = 1, )
• ( \lim_{h \to 0}\frac{\cos h - 1}{h} = 0 )

So:

[ \frac{d}{dx}\sin x = \sin x \cdot 0 + \cos x \cdot 1 = \cos x ]

The proof for ( \cos x ) is mechanically identical with ( \cos(x+h) ) in place of ( \sin(x+h) ). The other four derivatives — ( \tan, \cot, \sec, \csc ) — follow by writing each as a sine/cosine quotient and applying the quotient rule.

Quick derivation of ( \tan x )

[ \frac{d}{dx}\tan x = \frac{d}{dx}\left(\frac{\sin x}{\cos x}\right) = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \sec^2 x ]

The numerator collapses by the Pythagorean identity ( \sin^2 x + \cos^2 x = 1 ). That's why the trig derivatives are so tidy — Pythagoras is doing background work the whole time.

The Chain Rule for Trigonometric Functions

When the argument is a function ( u(x) ), each derivative picks up ( u^{\prime}(x) ):

[ \frac{d}{dx}\sin(u) = \cos(u) \cdot u^{\prime} \qquad \frac{d}{dx}\cos(u) = -\sin(u) \cdot u^{\prime} ]

[ \frac{d}{dx}\tan(u) = \sec^2(u) \cdot u^{\prime} \qquad \frac{d}{dx}\sec(u) = \sec(u)\tan(u) \cdot u^{\prime} ]

The chain rule is where most exam mistakes happen — and it's also where most of the action is. Almost every physics or engineering derivative involves ( \sin(\omega t) ) or ( \cos(\omega t) ) — the chain rule pulls the angular-frequency ( \omega ) out front.

Three Worked Examples — Quick, Standard, Stretch

Quick

Differentiate ( f(x)=3\sin x+2\cos x ) .

By linearity:

[ f^{\prime}(x)=3\cos x+2\cdot(-\sin x)=3\cos x-2\sin x ]

Done in one line. The negative sign on cosine is the only thing to watch.

Where Students Lose the Mark — A Worked Standard Example

Differentiate ( g(x)=\sin(3x^2) ) .

The wrong path. A student writes:

[ g^{\prime}(x)=\cos(3x^2) \text{❌} ]

They've remembered "the derivative of sine is cosine" and stopped there. The argument ( 3x^2 ) wasn't ( x ), so the chain rule applies — but it got skipped.

Sanity check. At ( x=0, ) this answer gives ( g^{\prime}(0)=\cos(0)=1 ). But ( g(x)=\sin(3x^2) ) is even (symmetric about the y-axis) — its derivative must be odd, and an odd function must satisfy ( g^{\prime}(0)=0 ). The answer 1 contradicts that. Something's missing.

The correct path. Apply the chain rule. Let ( u=3x^2, ) so ( u^{\prime}=6x ):

[ g^{\prime}(x)=\cos(3x^2)\cdot\frac{d}{dx}(3x^2)=6x\cos(3x^2) ]

Stretch

A particle's position at time ( t ) seconds is ( s(t)=4\sin(\frac{2\pi t}{5}) ) metres. Find its velocity and the maximum speed.

Velocity is ( s^{\prime}(t) ). With ( u=\frac{2\pi t}{5}, ) ( u^{\prime}=\frac{2\pi}{5} ):

[ v(t)=4\cdot\cos\left(\frac{2\pi t}{5}\right)\cdot\frac{2\pi}{5} = \frac{8\pi}{5}\cos\left(\frac{2\pi t}{5}\right) ]

Maximum speed is the amplitude of the cosine — the term in front:

[ v_{\max} = \frac{8\pi}{5} \approx 5.03 , \text{m/s} ]

Where These Derivatives Show Up in the Real World

The trig derivatives aren't just calculus furniture — they're how every oscillating system in physics gets analysed.

• Simple harmonic motion. Springs, pendulums, and tuning forks all satisfy ( s^{\prime\prime}=-\omega^2 s ) — and the only functions that solve that equation are ( \sin(\omega t) ) and ( \cos(\omega t) ). The reason the Tacoma Narrows Bridge collapsed in 1940 was that wind-induced vibrations matched the bridge's natural frequency — a calamity diagnosed afterward using exactly these derivatives.

• AC electrical circuits. Voltage in your wall outlet is ( V(t)=V_0\sin(2\pi\cdot60t) ) (in the US, McKinney TX included). Current through a capacitor is the derivative of voltage — a cosine wave. That phase shift of 90° between voltage and current is just ( \frac{d}{dt}\sin = \cos ).

• Sound and music. Every musical note is a superposition of sine waves. The derivative — the rate at which air pressure changes — is what your eardrum actually senses.

• Robotics and animation. When an animated character bobs up and down, the animator usually scripts position as a sine wave; the velocity (used for momentum-based blending) is the cosine derivative.

Tripping Points to Avoid

Four mistakes account for nearly every lost mark on this topic.

Mistake 1: Forgetting the negative sign on the "co-" derivatives

Where it slips in: Anywhere ( \cos, \cot, \text{or} \csc ) appears in a longer expression.

Mistake 2: Skipping the chain rule when the argument isn't ( x )

Where it slips in: Composite arguments like ( \sin(3x^2) ), ( \cos(x^3) ), ( \tan(\ln x) ) — exactly the Standard example above.

Mistake 3: Working in degrees instead of radians

Where it slips in: Calculator-heavy problems where the student forgets to switch the mode.

Mistake 4: Confusing the derivative of ( \sec x ) with ( \sec^2 x )

Where it slips in: Tangent and secant problems mixing up which one yields which derivative.

Key Takeaways

• The differentiation of trigonometric functions rests on ( \frac{d}{dx}\sin x = \cos x ) and ( \frac{d}{dx}\cos x = -\sin x ) — everything else is the quotient rule.
• "Co-" functions (cos, cot, csc) get a negative sign; the other three don't.
• The chain rule isn't optional when the argument is anything other than ( x ) alone.
• All rules assume radians — degrees introduce a ( \frac{\pi}{180} ) factor that's silently lost on calculators.

Try It Yourself — Three Problems

Differentiate the following without looking back at the table:

1. ( f(x)=\cos(5x) )
2. ( g(x)=\tan(x^2+1) )
3. ( h(x)=x\sin x ) (this one needs the product rule on top of the trig derivative)