# Derivatives of the Trigonometric Functions

Formulas of the derivatives of trigonometric functions sin(x), cos(x), tan(x), cot(x), sec(x) and csc(x), in calculus, are presented along with several examples involving products, sums and quotients of trigonometric functions.

## Formulae For The Derivatives of Trigonometric Functions

### 1 - Derivative of sin x

The derivative of \( f(x) = \sin x \) is given by

** \( f '(x) = \cos x \) **
### 2 - Derivative of cos x

The derivative of \( f(x) = \cos x \) is given by

** \( f '(x) = - \sin x \) **
### 3 - Derivative of tan x

The derivative of \( f(x) = \tan x \) is given by

** \( f '(x) = \sec^{2} x \) **
### 4 - Derivative of cot x

The derivative of \( f(x) = \cot x \) is given by

** \( f '(x) = - \csc^{2} x \) **
### 5 - Derivative of sec x

The derivative of \( f(x) = \sec x \) is given by

** \( f '(x) = \sec(x) \tan(x) \) **
### 6 - Derivative of csc x

The derivative of \( f(x) = \csc x \) is given by

** \( f '(x) = - \csc x \cot x \) **

## Examples Using the Derivatives of Trigonometric Functions

### Example 1

Find the first derivative of \( f(x) = x \sin x \)

__Solution to Example 1:__

- Let \( g(x) = x \) and \( h(x) = \sin x \), function \( f \) may be considered as the product of functions \( g \) and \( h \): \( f(x) = g(x) h(x) \). Hence we use the product rule, \( f '(x) = g(x) h '(x) + h(x) g '(x) \), to differentiate function \( f \) as follows

\( f '(x) = x \cos x + \sin x \cdot
1 = x \cos x + \sin x \)

### Example 2

Find the first derivative of \( f(x) = \tan x + \sec x \)

__Solution to Example 2:__

- Let \( g(x) = \tan x \) and \( h(x) = \sec x \), function \( f \) may be considered as the sum of functions \( g \) and \( h \): \( f(x) = g(x) + h(x) \). Hence we use the sum rule, \( f '(x) = g '(x) + h '(x) \), to differentiate function \( f \) as follows

\( f '(x) = \sec^{2} x + \sec x \tan x = \sec x (\sec x + \tan x) \)

### Example 3

Find the first derivative of \( f(x) = \dfrac{\sin x}{1 + \cos x} \)

__Solution to Example 3:__

- Let \( g(x) = \sin x \) and \( h(x) = 1 + \cos x \), function \( f \) may be considered as the quotient of functions \( g \) and \( h \): \( f(x) = \dfrac{g(x)}{h(x)} \). Hence we use the quotient rule, \( f '(x) = \dfrac{h(x) g '(x) - g(x) h '(x)}{h(x)^{2}} \), to differentiate function \( f \) as follows

\( g '(x) = \cos x \)

\( h '(x) = - \sin x \)

\( f '(x) = \dfrac{(1 + \cos x)(\cos x) - (\sin x)(- \sin x)}{(1 + \cos x)^{2}} \)

\( = \dfrac{\cos x + \cos^{2} x + \sin^{2} x}{(1 + \cos x)^{2}} \)

- Use trigonometric identity \( \cos^{2} x + \sin^{2} x = 1 \) to simplify the above

\( f '(x) = \dfrac{\cos x + 1}{(1 + \cos x)^{2}} = \dfrac{1}{\cos x + 1} \)

### More Links and References

Differentiation