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Calculus Differentiation

Derivatives of trig functions

20 practice questions 0 video lessons Theory + worked examples

Derivatives of Trig Functions

California Calculus • Standard 4.4 • Differentiation

Derivatives of Trig Functions is a topic in Differentiation in the California Calculus Standards. It is aligned to Standard 4.4, which requires students to derive and use the derivative formulas for trigonometric and inverse-trigonometric functions.

The trigonometric derivatives begin with \(\dfrac{d}{dx}\sin x=\cos x\) and \(\dfrac{d}{dx}\cos x=-\sin x\), and extend to \(\tan x\), the inverse-trig functions, and composites through the chain and product rules (angles in radians).

California Calculus › Differentiation › Derivatives of Trig Functions  —  Standard 4.4

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Theory

The trig derivatives start from \(\dfrac{d}{dx}\sin x=\cos x\) and \(\dfrac{d}{dx}\cos x=-\sin x\). With the chain rule and product rule you can differentiate any expression built from sine, cosine, and tangent. Angles are always in radians.

The three basic trigonometric derivatives are worth memorizing:

\[\dfrac{d}{dx}\sin x=\cos x,\quad\dfrac{d}{dx}\cos x=-\sin x,\quad\dfrac{d}{dx}\tan x=\sec^{2}x\]

Notice the minus sign on the derivative of cosine — it is the most common slip.

With the chain rule, a constant inside comes out as a factor: \(\dfrac{d}{dx}\sin(kx)=k\cos(kx)\).

Key idea: these formulas only hold in radians. They come directly from the special limit \(\dfrac{\sin x}{x}\to 1\).
The sine curve and its derivative the cosine curve The derivative of sin x is cos x; where sin has a peak the cosine crosses zero. x y sin x cos x
\(\dfrac{d}{dx}\sin x=\cos x\): the sine and its derivative.
A tangent to the sine curve whose slope is cos of the point The slope of the tangent to sin x at any point equals the cosine of that point. x y
The slope of \(\sin x\) at a point equals \(\cos\) of that point.

The core derivatives and one inverse:

\[\dfrac{d}{dx}\sin x=\cos x,\qquad\dfrac{d}{dx}\cos x=-\sin x\]

derivative of sine is cosine; derivative of cosine is negative sine

\[\dfrac{d}{dx}\tan x=\sec^{2}x,\qquad\dfrac{d}{dx}\arctan x=\dfrac{1}{1+x^{2}}\]

derivative of tangent is secant squared; derivative of arctangent is one over one plus x squared

With the chain rule: \(\dfrac{d}{dx}\sin(g(x))=\cos(g(x))\cdot g'(x)\), and similarly for cosine and tangent.

How to differentiate a trig expression

  1. Identify the outer trig function and its inside.
  2. Apply the basic derivative (mind the minus sign on cosine), keeping the inside unchanged.
  3. Multiply by the inner derivative (chain rule); use the product rule if \(x\) multiplies a trig factor.
Example 1 — Sum of trig functions
Differentiate \(y=\sin x+\cos x\).
Solution

Use \(\dfrac{d}{dx}\sin x=\cos x\) and \(\dfrac{d}{dx}\cos x=-\sin x\).

\(y'\)\(=\)\(\cos x-\sin x\)

derivative is cos x minus sin x

Example 2 — Chain rule with sine
Differentiate \(y=\sin 3x\).
Solution

Outer \(\sin\), inner \(3x\) with derivative \(3\).

\(y'\)\(=\)\(\cos(3x)\cdot 3\)
\(=\)\(3\cos 3x\)

derivative is 3 cos 3x

Example 3 — Product with a power
Differentiate \(y=x^2\cos x\).
Solution

Product rule with \(u=x^2\), \(v=\cos x\).

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

derivative is 2x cos x minus x squared sin x

Example 4 — Tangent and inverse tangent
Differentiate \(y=\tan x\) and \(y=\arctan x\).
Solution

These are the two named trig-derivative facts.

\(\dfrac{d}{dx}\tan x\)\(=\)\(\sec^2 x\)
\(\dfrac{d}{dx}\arctan x\)\(=\)\(\dfrac{1}{1+x^2}\)

derivative of tan is sec squared; derivative of arctan is one over one plus x squared

Common pitfalls

The minus sign on cosine. \(\dfrac{d}{dx}\cos x=-\sin x\); forgetting the sign is the classic error.
Chain-rule factor for \(\sin(kx)\). The derivative is \(k\cos(kx)\); the constant \(k\) must appear.
Work in radians. These derivative formulas are false for degree measure.

Frequently asked questions

What is the derivative of sin x?

\(\dfrac{d}{dx}\sin x=\cos x\), with \(x\) in radians.

What is the derivative of cos x?

\(\dfrac{d}{dx}\cos x=-\sin x\). Note the minus sign.

What is the derivative of tan x?

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

How do you differentiate sin(3x)?

Use the chain rule: \(\dfrac{d}{dx}\sin 3x=3\cos 3x\). The inner derivative \(3\) comes out front.

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