Differentiation of Hyperbolic Functions
Formulas and examples, with detailed solutions, on the derivatives of hyperbolic functions are presented. For definitions and graphs of hyperbolic functions go to Graphs of Hyperbolic Functions

function

derivative

f(x) = sinh x

f '(x) = cosh x

f(x) = cosh x

f '(x) = sinh x

f(x) = tanh x
= sinh x / cosh x 
f '(x) = sech^{ 2} x

f(x) = coth x = 1 / tanh x
= cosh x / sinh x 
f '(x) =  csch ^{ 2}x

f(x) = csch x
= 1 / sinh x

f '(x) =  csch x coth x

f(x) = sech x
= 1 / cosh x

f '(x) =  sech x tanh x

Example 1: Find the derivative of f(x) = sinh (x^{ 2})
Solution to Example 1:

Let u = x^{ 2} and y = sinh u and use the chain rule to find the derivative of the given function f as follows.
f '(x) = (dy / du) (du / dx)

dy / du = cosh u, see formula above, and du / dx = 2 x
f '(x) = 2 x cosh u = 2 x cosh (x^{ 2})

Substitute u = x^{ 2} in f '(x) to obtain
f '(x) = 2 x cosh (x^{ 2})
Example 2: Find the derivative of f(x) = 2 sinh x + 4 cosh x
Solution to Example 2:

Let g(x) = 2 sinh x and h(x) = 4 cosh x, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f
f '(x) = 2 cosh x + 4 sinh x
Example 3: Find the derivative of f(x) = cosh x / sinh (x^{ 2})
Solution to Example 3:

Let g(x) = cosh x and h(x) = sinh x^{ 2}, function f is 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 find the derivative of function f.
g '(x) = sinh x
h '(x) = 2 x cosh x^{ 2} (see example 2 above)
f '(x) = [ h(x) g '(x)  g(x) h '(x) ] / h(x)^{ 2}
= [ (sinh x^{ 2}) (sinh x)  (cosh x)(2 x cosh x^{ 2}) ] / (sinh x^{ 2})^{ 2}
Example 4: Find the derivative of f(x) = (sinh x)^{ 2}
Solution to Example 4:

Let u = sinh x and y = u^{ 2}, Use the chain rule to find the derivative of function f as follows.
f '(x) = (dy / du) (du / dx)

dy / du = 2 u and du / dx = cosh x
f '(x) = 2 u cosh x

Put u = sinh x in f '(x) obtained above
f '(x) = 2 sinh x cosh x
Exercises Find the derivative of each function.
1  f(x) = sinh (x^{ 3})
2  g(x) =  sinh x + 4 cosh (x + 2)
3  h(x) = cosh x^{ 2}/ sinh x
4  j(x) =  (cosh x)^{ 2}
solutions to the above exercises
1  f '(x) = (3x^{ 2}) cosh (x^{ 3})
2  g '(x) =  cosh x + 4 sinh (x + 2)
3  h '(x) = [ (2 x sinh x^{ 2})(sinh x)  (cosh x^{ 2})(cosh x) ] / [sinh x]^{ 2}
4  j '(x) =  2 (cosh x)(sinh x)
More on differentiation and derivatives  
Linear ProgrammingNew !
Online Step by Step Calculus Calculators and SolversNew !
Factor Quadratic Expressions  Step by Step CalculatorNew !
Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers

Interactive HTML5 Math Web Apps for Mobile LearningNew !

Free Online Graph Plotter for All Devices
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: February 2015
Copyright © 2003  2015  All rights reserved