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

2 - Derivative of cos x

3 - Derivative of tan x

4 - Derivative of cot x

5 - Derivative of sec x

6 - Derivative of csc x

Examples Using the Derivatives of Trigonometric Functions

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

• 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

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

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