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Mathematics Learning Centre

Derivatives of trigonometric

functions

Christopher Thomas

University of Sydney

1 Derivatives of trigonometric functions

To understand this section properly you will need to know about trigonometric functions. The Mathematics Learning Centre booklet Introduction to Trigonometric Functions may be of use to you.

There are only two basic rules for differentiating trigonometric functions:

d dx sin x = cos x d dx cos x = − sin x.

For differentiating all trigonometric functions these are the only two things that we need to remember.

Of course all the rules that we have already learnt still work with the trigonometric functions. Thus we can use the product, quotient and chain rules to differentiate functions that are combinations of the trigonometric functions.

For example, tan x = (^) cossin^ xx and so we can use the quotient rule to calculate the derivative.

f (x) = tan x = sin x cos x

f ′(x) = cos x.(cos x) − sin x.(− sin x) (cos x)^2

= cos^2 x + sin^2 x cos x

cos^2 x (since cos^2 x + sin^2 x = 1) = sec^2 x

Note also that

cos^2 x + sin^2 x cos^2 x

cos^2 x cos^2 x

sin^2 x cos^2 x = 1 + tan^2 x

so it is also true that

d dx tan x = sec^2 x = 1 + tan^2 x.

By the quotient rule

d cot x dx

− sin^2 x − cos^2 x sin^2 x

sin^2 x

Using the composite function rule

d sec x dx

d(cos x)−^1 dx = −(cos x)−^2 × (− sin x) = sin x cos^2 x

d csc x dx

d(sin x)−^1 dx = −(sin x)−^2 × cos x = − cos x sin^2 x

Exercise 1

Differentiate the following:

a. cos 3x b. sin(4x + 5) c. sin^3 x d. sin x cos x e. x^2 sin x

f. cos(x^2 + 1) g. sin x x h. sin

x i. tan(

x) j.

x sin

x

Solutions to Exercise 1

a. d dx cos 3x = −3 sin 3x

b. d dx sin(4x + 5) = 4 cos(4x + 5)

c. d dx

sin^3 x = 3 sin^2 x cos x

d.

d dx sin x cos x = cos^2 x − sin^2 x

e. d dx x^2 sin x = 2x sin x + x^2 cos x

f. d dx cos(x^2 + 1) = − 2 x sin(x^2 + 1)

g. d dx

( (^) sin x x

)

x cos x − sin x x^2

h. d dx sin

x

x^2 cos

x

i. d dx tan

x =

x sec^2

x

j. d dx

x sin

x

) = −

x^2 sin

x

x^3 cos

x