Derivatives of Trigonometric Functions

What You’ll Learn

In this lesson you’ll learn the derivatives of all six trig functions and how to use them with the chain rule, product rule, and quotient rule.

The Concept

You already know the derivatives of polynomials. Now we add trig functions to the toolkit. There are six to memorize, but they come in pairs and the pattern is consistent.

The big two:

$$\frac{d}{dx}[\sin x] = \cos x$$ $$\frac{d}{dx}[\cos x] = -\sin x$$

Notice the negative sign on cosine’s derivative. That’s the one people forget.

The tangent and cotangent pair:

$$\frac{d}{dx}[\tan x] = \sec^2 x$$ $$\frac{d}{dx}[\cot x] = -\csc^2 x$$

The secant and cosecant pair:

$$\frac{d}{dx}[\sec x] = \sec x \tan x$$ $$\frac{d}{dx}[\csc x] = -\csc x \cot x$$

The pattern: the “co” functions (cos, cot, csc) all pick up a negative sign. If you remember that, you only need to memorize three derivatives and negate them for the co-versions.

These almost always show up with the chain rule. When the argument isn’t just x (like sin(3x²) or cos(5t + 1)), you differentiate the trig part first, then multiply by the derivative of the inside.

Worked Example

Example 1: Chain rule with sine

Find the derivative of f(x) = sin(4x).

The outside is sin(u), derivative is cos(u). The inside is 4x, derivative is 4.

$$f'(x) = \cos(4x) \cdot 4 = 4\cos(4x)$$

Example 2: Product rule with cosine

Differentiate y = x² cos(x).

Let u = x², u’ = 2x. Let v = cos(x), v’ = -sin(x).

$$y' = 2x\cos(x) + x^2(-\sin x) = 2x\cos x - x^2\sin x$$

Example 3: Chain rule with tangent

Find the derivative of y = tan(3x² + 1).

Outside: tan(u), derivative is sec²(u). Inside: 3x² + 1, derivative is 6x.

$$y' = \sec^2(3x^2 + 1) \cdot 6x = 6x\sec^2(3x^2 + 1)$$

Real-World Application

Trig derivatives are everywhere in physics and engineering:

• Simple harmonic motion: a mass on a spring has position s(t) = A sin(wt). The velocity is s’(t) = Aw cos(wt) and acceleration is s”(t) = -Aw² sin(wt). The derivatives cycle between sin and cos.
• Waves: sound, light, and water waves are all modeled with sin and cos. Their derivatives describe how fast the wave is changing at each point.
• Electrical engineering: AC voltage is V(t) = V₀ sin(wt). The rate of voltage change is V’(t) = V₀w cos(wt).
• Pendulums, vibrations, and anything that oscillates uses these derivatives.

You’ve Got This

Six derivatives, but really just three patterns with a negative sign rule for the “co” functions. The derivative of sin is cos. The derivative of cos is negative sin. The derivative of tan is sec². Everything else follows the same logic. Once you’ve done a few problems combining these with the chain rule, the pattern becomes second nature.

Quiz

The derivative of cos(x) is:

• A.sin x
• B.-sin x
• C.cos x
• D.-cos x

The derivative of sin(3x) is:

• A.cos(3x)
• B.-3sin(3x)
• C.3cos(3x)
• D.3sin(3x)

The derivative of tan(x) is:

• A.sec x tan x
• B.-csc² x
• C.cos² x
• D.sec² x

Which trig derivatives have a negative sign?

• A.sin, tan, sec
• B.All six of them
• C.cos, cot, csc (the co-functions)
• D.None of them

The derivative of sec(x) is:

• A.sec² x
• B.-csc x cot x
• C.sec x tan x
• D.tan² x