Differentiation of 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.
 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 ^{ 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) = [ cos x + 1 ] / (1 + cos x)^{ 2} = 1 / [cos x + 1]
More references on
Differentiation  
Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: 2 April 2013
Copyright © 2003  2014  All rights reserved