On this page we discuss the derivative of hyperbolic functions. First, we give a quick precalculus review then we discuss the basics of the derivatives.
Precalculus Review  What Are Hyperbolic Functions?
Hyperbolic functions are functions formed from exponentials. They appear so often that they are given the special name hyperbolic and they seem to work similar to trig functions, so they are also called hyperbolic trig functions. In trigonometry we have sine, cosine, tangent, etc. Hyperbolic functions are called hyperbolic sine, hyperbolic cosine, hyperbolic tangent and the abbreviations are written \(\sinh(x), \cosh(x), \tanh(x) . . . \), with an 'h' after the same name as the trig functions. Here are the definitions.
\(\displaystyle{ \sinh(x) = \frac{e^xe^{x}}{2} }\) 
\(\displaystyle{ \cosh(x) = \frac{e^x+e^{x}}{2} }\)  
\(\displaystyle{ \csch(x) = \frac{1}{\sinh(x)} }\) 
\(\displaystyle{ \sech(x) = \frac{1}{\cosh(x)} }\)  
\(\displaystyle{ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} }\) 
\(\displaystyle{ \coth(x) = \frac{\cosh(x)}{\sinh(x)} }\) 
So, you can see why we call them hyperbolic trig functions. Other than \(\sinh(x)\) and \(\cosh(x)\), we define the other hyperbolic functions similar to the corresponding definitions for trig functions. Now let's look at some identities and I think the relationship will become clearer.
\(\displaystyle{ \sinh(x) = \sinh(x) }\) 
\(\displaystyle{ \cosh(x) = \cosh(x) }\)  
\(\displaystyle{ \cosh^2(x)  \sinh^2(x) = 1 }\) 
\(\displaystyle{ 1\tanh^2(x) = \sech^2(x) }\) 
Be careful here. Notice that although they are similar, they are NOT always identical, usually with a change in sign or some other subtle difference. So don't gloss over this. Take a few minutes to notice the similarities AND the differences. Before we get started with the equations, let's take a few minutes to watch this video that explains hyperbolic functions using graphs. I think this will help you get a good feel for them.
video by Michel vanBiezen 

Let's start with a very good video discussing some of these equations and working with them to prove some identities. This is a great video to get you started with hyperbolic functions.
video by PatrickJMT 

Okay, let's take a look at another quick video working with hyperbolic functions. Using the definitions for \(\cosh(x)\) and \(\sinh(x)\), this presenter proves the identity \(\displaystyle{ \cosh^2(x)  \sinh^2(x) = 1 }\) . Although the end result is not particularly significant, this video shows how you can work with hyperbolic trig functions using the definitions.
video by Krista King Math 

Remember, there is nothing special or magic going on here. We just define \(\sinh(x)\) and \(\cosh(x)\) using exponentials and everything else builds from there.
Derivatives of Hyperbolic Functions
Okay, since nothing special is going on, you should be able to determine the derivatives of each hyperbolic function based only on exponentials. Doing so, produces the following formulas.
derivatives 

\(\displaystyle{ \frac{d}{dx}[\sinh(u)] = \cosh(u)\frac{du}{dx} }\) 
\(\displaystyle{ \frac{d}{dx}[\cosh(u)] = \sinh(u)\frac{du}{dx} }\) 
\(\displaystyle{ \frac{d}{dx}[\tanh(u)] = \sech^2(u)\frac{du}{dx} }\) 
\(\displaystyle{ \frac{d}{dx}[\coth(u)] = \csch^2(u)\frac{du}{dx} }\) 
\(\displaystyle{ \frac{d}{dx}[\sech(u)] = }\) \(\displaystyle{ \sech(u)\tanh(u)\frac{du}{dx} }\) 
\(\displaystyle{ \frac{d}{dx}[\csch(u)] = }\) \(\displaystyle{ \csch(u)\coth(u)\frac{du}{dx} }\) 
Again, notice similarities but also differences (especially minus signs) between these equations and the corresponding trig function derivatives.
Here is a video clip explaining some of these derivatives and showing how to use the exponential definitions to calculate the derivative.
video by PatrickJMT 

Okay, let's try some practice problems.
Practice
Unless otherwise instructed, calculate the derivative of these functions. Give your final answers in exact, completely factored form.
\(f(x)=\tanh(4x)\)
Problem Statement 

Calculate the derivative of the function \(f(x)=\tanh(4x)\). Give your final answer in exact, completely factored form.
Solution 

video by PatrickJMT 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(f(x)=\ln(\sinh(x))\)
Problem Statement 

Calculate the derivative of the function \(f(x)=\ln(\sinh(x))\). Give your final answer in exact, completely factored form.
Solution 

video by PatrickJMT 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(f(x)=\sinh(x)\tanh(x)\)
Problem Statement 

Calculate the derivative of the function \(f(x)=\sinh(x)\tanh(x)\). Give your final answer in exact, completely factored form.
Solution 

video by PatrickJMT 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(f(x)=\text{sech}^2(e^x)\)
Problem Statement 

Calculate the derivative of the function \(f(x)=\text{sech}^2(e^x)\). Give your final answer in exact, completely factored form.
Solution 

video by Krista King Math 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(f(x)=\sin(\sinh(x))\)
Problem Statement 

Calculate the derivative of the function \(f(x)=\sin(\sinh(x))\). Give your final answer in exact, completely factored form.
Solution 

video by Krista King Math 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(f(x)=\sin\left[ \sinh(3x^2+2x) \right]\)
Problem Statement 

Calculate the derivative of the function \(f(x)=\sin\left[ \sinh(3x^2+2x) \right]\). Give your final answer in exact, completely factored form.
Solution 

video by Krista King Math 

close solution

Log in to rate this practice problem and to see it's current rating. 

You CAN Ace Calculus
external links you may find helpful 

To bookmark this page and practice problems, log in to your account or set up a free account.
Single Variable Calculus 

MultiVariable Calculus 

Differential Equations 

Precalculus 

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
free ideas to save on books 

Help Keep 17Calculus Free 

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. 
How to Develop a Brilliant Memory Week by Week: 50 Proven Ways to Enhance Your Memory Skills 

Practice Instructions
Unless otherwise instructed, calculate the derivative of these functions. Give your final answers in exact, completely factored form.