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Differentiation – Derivatives of trigonometric functions

📘 Notes

Differentiation — Derivatives of Trigonometric Functions

The derivatives of trigonometric functions are important rules in Year 13 calculus.

1. Basic derivatives

The main trigonometric derivatives are:

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

These rules are for angles measured in radians.

2. Important note about radians

These derivative formulas only work in the standard form when angles are in radians.

\[ \text{Use radians, not degrees.} \]

3. Using the chain rule

If the angle is a function of \(x\), then use the chain rule.

\[ \frac{d}{dx}(\sin(f(x)))=f'(x)\cos(f(x)) \] \[ \frac{d}{dx}(\cos(f(x)))=-f'(x)\sin(f(x)) \] \[ \frac{d}{dx}(\tan(f(x)))=f'(x)\sec^2(f(x)) \]

Differentiate the outside trig function, then multiply by the derivative of the inside.

4. Worked example 1

Differentiate

\[ y=\sin x \]

\[ \frac{dy}{dx}=\cos x \]

5. Worked example 2

Differentiate

\[ y=\cos x \]

\[ \frac{dy}{dx}=-\sin x \]

6. Worked example 3

Differentiate

\[ y=\tan x \]

\[ \frac{dy}{dx}=\sec^2 x \]

7. Worked example 4

Differentiate

\[ y=\sin(3x) \]

Use the chain rule. The derivative of \(3x\) is \(3\).

\[ \frac{dy}{dx}=3\cos(3x) \]

8. Worked example 5

Differentiate

\[ y=\cos(x^2+1) \]

The derivative of the inside \(x^2+1\) is \(2x\).

\[ \frac{dy}{dx}=-2x\sin(x^2+1) \]

9. Worked example 6

Differentiate

\[ y=\tan(2x-5) \]

The derivative of the inside \(2x-5\) is \(2\).

\[ \frac{dy}{dx}=2\sec^2(2x-5) \]

10. Trigonometric functions with product or quotient rules

Trigonometric functions are often combined with other functions.

Example: Differentiate

\[ y=x\sin x \]

Use the product rule:

\[ \frac{dy}{dx}=x\cos x+\sin x \]

Here both the product rule and trig derivatives are needed.

11. Common mistakes

  • Forgetting that \(\dfrac{d}{dx}(\cos x)=-\sin x\).
  • Forgetting the chain rule for \(\sin(3x)\), \(\cos(x^2)\), etc.
  • Writing \(\dfrac{d}{dx}(\tan x)=\tan^2 x\), which is incorrect.
  • Using degrees instead of radians.

12. Exam tips

  • Memorise: \[ \frac{d}{dx}(\sin x)=\cos x,\quad \frac{d}{dx}(\cos x)=-\sin x,\quad \frac{d}{dx}(\tan x)=\sec^2 x \]
  • Always check whether the chain rule is needed.
  • Keep the negative sign with the derivative of \(\cos x\).
  • Look out for product rule or quotient rule as well.
  • Make sure angles are in radians.