## Differentiation of Trigonometric Functions Each of the basic trigonometric functions can be differentiated.

### Derivatives of sin x, cos x and tan x

 If f(x) = sin x then f '(x) = cos x proof If f(x) = cos x then f '(x) = − sin x If f(x) = tan x then f '(x) = sec2x proof

Examples using Chain Rule

 Differentiate f(x) = sin 4x f '(x) = 4 cos 4x Differentiate f(x) = 4 cos 2x f '(x) = 2. 4. -sin 2x         = -8 sin 2x

### Derivatives of sec x, cosec x and cot x

 If f(x) = sec x then f '(x) = sec x tan x proof If f(x) = cosec x then f '(x) = − cosec x cot x proof If f(x) = cot x then f '(x) = − cosec2x proof

Examples using Chain Rule

 Differentiate f(x) = cot (3x + 2) f '(x) = 3. -cosec2 (3x + 2)          = -3cosec2 (3x + 2) Differentiate f(x) = sec 4x f '(x) = 4 sec 4x tan 4x

### Derivatives of the Inverse Trigonometric Functions proof proof proof

For more complex inverse trig. functions the following can be used. proof proof proof

These formulae are often provided for tests and examinations. Make sure you check with your teacher whether they are provided.

Examples

1. Differentiate y = sin-1(2x + 1)

Let u = 2x + 1 ∴ y = sin-1u 2. Differentiate y = x tan-1x 3. Differentiate y = sec-1x = cos-1 1/x 