Differentiating Trigonometric Functions
Differentiating trigonometric functions uses \(\dfrac{d}{dx}\sin x=\cos x\), \(\dfrac{d}{dx}\cos x=-\sin x\) and \(\dfrac{d}{dx}\tan x=\sec^{2}x\) (radians).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Differential calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
Sine differentiates to cosine, and cosine to negative sine (in radians). This Year 12 Mathematics Advanced topic (MAV-12-04) differentiates \(\sin\), \(\cos\) and \(\tan\) with the chain rule.
\(\dfrac{d}{dx}\sin x=\cos x\), \(\dfrac{d}{dx}\cos x=-\sin x\), \(\dfrac{d}{dx}\tan x=\sec^{2}x\). With the chain rule, \(\dfrac{d}{dx}\sin(ax)=a\cos(ax)\), and similarly for the others.
These hold only when \(x\) is in radians.
Method
- Recall the base derivative (sin, cos or tan).
- Multiply by the derivative of the inner angle (chain rule).
- Work in radians.
| \(\dfrac{dy}{dx}\) | \(=\) | \(3\cos 3x\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(-2x\sin(x^{2})\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\sec^{2}x\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(2\cos x\) |
| \(\text{at }x=0\) | \(=\) | \(2\) |
Common pitfalls
Frequently asked questions
What is the derivative of sin x?
It is cos x, when x is in radians.
What is the derivative of cos x?
It is minus sin x.
What is the derivative of tan x?
It is sec squared x.
Why must angles be in radians for these derivatives?
The simple rules like the derivative of sin x being cos x only hold when x is measured in radians; in degrees an extra constant factor appears.