Differentiating Trigonometric Functions
Learn differentiating trigonometric functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-04). Working in radians, the gradient of the sine curve follows the cosine curve, and the gradient of cosine follows negative sine — results you can see directly from the graphs.
You will learn the derivatives of \(\sin x\), \(\cos x\) and \(\tan x\), find those of sec, cosec and cot, and use the chain rule for forms like \(\sin(3x)\) or \(\cos(x^2)\) — essential for any problem involving waves, rotation or other cycles.
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.