Differentiation of Trigonometric Functions

Formulas of the derivatives of trigonometric functions, in calculus, are presented along with several examples involving products, sums and quotients 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 tan x is given by

f '(x) = sec x tan x


6 - Derivative of csc x.

The derivative of f(x) = csc xis given by

f '(x) = - csc x cot x


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 * 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) = 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) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ 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) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2

    = [ (1 + cos x)(cos x) - (sin x)(- sin x) ] / (1 + cos x) 2

    = [ cos x + cos 2x + sin 2x ] / (1 + cos x) 2

  • Use trigonometric identity cos 2x + sin 2x = 1 to simplify the above

    f '(x) = [ cos x + 1 ] / (1 + cos x) 2 = 1 / [cos x + 1]


More references on Differentiation


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Updated: 3 April 2011

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