The basic hyperbolic functions are the hyperbolic sine function and the hyperbolic cosine function. They are defined as follows:. The derivatives of hyperbolic functions can be easily found as these functions are defined in terms of exponential functions.
So, the derivatives of the hyperbolic sine and hyperbolic cosine functions are given by. We can easily obtain the derivative formula for the hyperbolic tangent :. It is known that the hyperbolic sine and cosine are connected by the relationship.
Similarly, we can find the differentiation formulas for the other hyperbolic functions:. As you can see, the derivatives of the hyperbolic functions are very similar to the derivatives of trigonometric functions. However, it is important to note the difference in signs!
If the derivative of the cosine function is given by. For the secant function , the situation with the sign is exactly reversed:.
The corresponding differentiation formulas can be derived using the inverse function theorem. Then the derivative of the inverse hyperbolic sine is given by.
Similarly, we can obtain the derivatives for the inverse hyperbolic cosine , tangent and cotangent functions. We also derive the derivatives of the inverse hyperbolic secant and cosecant , though these functions are rare.
List of Derivatives of Hyperbolic & Inverse Hyperbolic Functions
Similarly, we can find the derivative of the inverse hyperbolic cosecant. The derivative of the inverse hyperbolic cosecant is expressed as.
By combining the two branches of the solutions, we obtain the final expression for the derivative of the inverse hyperbolic cosecant in the form.
For convenience, we collect the differentiation formulas for all hyperbolic functions in one table:.
Using the chain rule , we obtain:. Using the difference and product rule , we have:. Example 1.
Differentiating hyperbolic functions (short exercise) : ExamSolutions
Example 2. Example 3. Example 4.
1,001 Chemistry Practice Problems For Dummies
Example 5. Example 6. Page 1.
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