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.
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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.
\(\displaystyle{ \sinh(x) = \frac{e^x-e^{-x}}{2} }\) |
\(\displaystyle{ \cosh(x) = \frac{e^x+e^{-x}}{2} }\) | |
\(\displaystyle{ \csch(x) = \frac{1}{\sinh(x)} }\) |
\(\displaystyle{ \sech(x) = \frac{1}{\cosh(x)} }\) | |
\(\displaystyle{ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} }\) |
\(\displaystyle{ \coth(x) = \frac{\cosh(x)}{\sinh(x)} }\) |
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 |
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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 |
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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 |
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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 |
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\(\displaystyle{ \frac{d}{dx}[\sinh(u)] = \cosh(u)\frac{du}{dx} }\) |
\(\displaystyle{ \frac{d}{dx}[\cosh(u)] = \sinh(u)\frac{du}{dx} }\) |
\(\displaystyle{ \frac{d}{dx}[\tanh(u)] = \sech^2(u)\frac{du}{dx} }\) |
\(\displaystyle{ \frac{d}{dx}[\coth(u)] = -\csch^2(u)\frac{du}{dx} }\) |
\(\displaystyle{ \frac{d}{dx}[\sech(u)] = }\) \(\displaystyle{ -\sech(u)\tanh(u)\frac{du}{dx} }\) |
\(\displaystyle{ \frac{d}{dx}[\csch(u)] = }\) \(\displaystyle{ -\csch(u)\coth(u)\frac{du}{dx} }\) |
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 |
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Okay, let's try some practice problems.
Practice
Unless otherwise instructed, calculate the derivative of these functions. Give your final answers in exact, simplified, 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 |
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\(f(x)=\ln(\sinh(x))\)
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 |
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\(f(x)=\sinh(x)\tanh(x)\)
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 |
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\(f(x)=\text{sech}^2(e^x)\)
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 |
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\(f(x)=\sin(\sinh(x))\)
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 |
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\(f(x)=\sin\left[ \sinh(3x^2+2x) \right]\)
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 |
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