What are hyperbolic trig functions?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Great question!
And that’s exactly what you’re going to learn about in today’s calculus class.
Let’s go!
Background
Did you know that specific combinations of exponential function arise so frequently in mathematical and physical situations that they are given the special name of hyperbolic functions?
Not only do these combinations share a striking resemblance to the hyperbola, but they are in many ways equivalent to their cousins the circular trigonometric functions, as they share many similarities in their:
- Graphs
- Identities
- Derivatives
- Integral Properties
Defined
The definition of the Hyperbolic Functions, along with their identities, is displayed in the chart below:
Hyperbolic Trig Identities
And the graphs for hyperbolic sine, hyperbolic cosine, and hyperbolic tangent are depicted as follows:
Graph Of Sinhx
Graph Of Coshx
Graph Of Tanhx
Derivative Of Hyperbolic Functions
And the derivatives of the hyperbolic trig functions are easily computed, and you will undoubtedly see the similarities to the well-known trigonometric derivatives.
Notice, however, that some of the signs are different, as noted by Whitman College. In particular, sinh, cosh, and tanh, or as I like to refer to them as the “big three,” are all positive.
Derivatives Of Hyperbolic Functions
Sinh Proof
Now before we look at a few problems, I want to take a moment to walk through the steps for proving the differentiation rule for y= sinh(x), as the steps shown below are similar to how we would prove the rest.
Prove Sinhx Equals Coshx
But thankfully, we don’t need to use the definitions or identities of hyperbolic functions to compute derivatives. All we have to do is use our differentiation formulas.
Examples – Differentiating
Okay, so let’s look at how we can easily apply our differentiation formulas for hyperbolic trig functions with a few examples. We know all of our derivative techniques. It’s now time to talk about taking the derivatives of hyperbolic functions.
Hyperbolic Functions Examples
See how easy these rules are!
Inverse Hyperbolic Trig Derivatives
And just as trigonometric functions can be expressed as inverses, hyperbolic trig functions can similarly be defined.
Inverse Hyperbolic Trig Derivatives
Again, you will notice how strikingly similar the inverse trig and inverse hyperbolic trig derivates are, just with a slight sign change.
Examples – Inverse Differentiation
So, let’s look at a few examples of how we take inverse hyperbolic trig derivatives.
Inverse Hyperbolic Functions Examples
Summary
So, all in all, we just have to plug into our formulas and simplify!
Together we will use our new differentiation rules for hyperbolic trigonometric functions combined with our other important derivative formulas and skills for polynomials, exponentials, and logarithmic functions too!
Let’s jump right in!
Video Tutorial w/ Full Lesson & Detailed Examples (Video)
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