Derivative of Trigonometric Function

3.3 Derivatives of Trigonometric Functions
Math 1271, TA: Amy DeCelles

1. Overview
You need to memorize the derivatives of all the trigonometric functions. If you don’t get them straight before we learn integration, it will be much harder to remember them correctly.
(sin x)

=

cos x

(cos x)

=

− sin x

(tan x)

=

sec2 x

(sec x)

=

sec x tan x

(csc x)

=

− csc x cot x

(cot x)

=

− csc2 x

A couple of useful limits also appear in this section: lim θ→0

lim

θ→0

sin θ
=1
θ

cos θ − 1
=0
θ

2. Examples
1.) Find the derivative of g(x) = 4 sec t + tan t
We use the derivatives of sec and tan: g (x) = 4 sec t tan t + sec2 t
2.) Find the derivative of
1 + sin x x + cos x
Since y is the quotient of two functions we first use the quotient rule: y= y =

(1 + sin x) (x + cos x) − (1 + sin x)(x + cos x)
(x + cos x)2

Evaluating the derivatives we get: y =

(cos x)(x + cos x) − (1 + sin x)(1 − sin x)
(x + cos x)2

Simplifying the numerator: y =

(x cos x + cos2 x) − (1 − sin2 x)
(x + cos x)2

=

x cos x + cos2 x − 1 + sin2 x
(x + cos x)2

We now use the trig identity sin2 + cos2 = 1: y =

x cos x x cos x − 1 + 1
=
2
(x + cos x)
(x + cos x)2

2.) Find the derivative of y = x sin x cos x
Since y is a product of functions we’ll use the product rule. We have to use it twice, actually, because y is a product of three functions. Applying it once, we get: y = (x) (sin x cos x) + (x)(sin x cos x)
And now applying it to the product sin x cos x, we get: y = (x) (sin x cos x) + (x)((sin x) (cos x) + (sin x)(cos x) )
Now taking derivatives, we get: y = sin x cos x + x((cos x)(cos x) + (sin x)(− sin x))
And simplifying: y = sin x cos x + x(cos2 x − sin2 x) = sin x cos x + x cos2 x − x sin2 x

3.) Find tangent to the curve at the point (0, 1). y= 1 sin x + cos x

The slope of the tangent line will be the value of the derivative at x = 0. So the first
thing