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
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:
\[ f'(x) = \dfrac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} \]We find the individual derivatives:
\[ g'(x) = \cos x \] \[ h'(x) = -\sin x \]Substitute into the quotient rule formula:
\[ f'(x) = \dfrac{(1 + \cos x)(\cos x) - (\sin x)(-\sin x)}{(1 + \cos x)^2} \] \[ = \dfrac{\cos x + \cos^2 x + \sin^2 x}{(1 + \cos x)^2} \]Use the trigonometric identity \( \cos^2 x + \sin^2 x = 1 \) to simplify:
\[ f'(x) = \dfrac{\cos x + 1}{(1 + \cos x)^2} = \dfrac{1}{\cos x + 1} \]