Calculus / Unit 4: Differentiation Rules
4.5 Derivatives of Trigonometric Functions
Notes
Memorize these (with $x$ in radians).
| Function | Derivative |
|---|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
- With the chain rule: $\displaystyle \frac{d}{dx}[\sin(2x)] = 2\cos(2x)$.
- Combine with the other rules: $\displaystyle \frac{d}{dx}[x \cos x] = \cos x - x \sin x$.
- The derivatives of the "co" functions have a negative sign.
Practice
Find $\displaystyle \frac{d}{dx}[3\sin x + \cos x]$.
$(\sin x)' = \cos x$ and $(\cos x)' = -\sin x$.
Find $\displaystyle \frac{d}{dx}[\sin(2x)]$.
Outside: $\cos(2x)$. Inside derivative: 2.
Find $\displaystyle \frac{d}{dx}[x\cos x]$.
Product rule: $(1)\cos x + x(-\sin x)$.
Ready for more? Take the Unit 4 test.