Introduction to Hyperbolic Differentiation
There are six basic hyperbolic functions in geometry. The rate of change of these functions is known as hyperbolic differentiation. This method follows the derivative rules and formulas to calculate the rate of change. Let’s understand how to apply hyper differentiation step-by-step and learn the difference between hyperbolic and trig differentiation.
Understanding the Hyperbolic Differentiation
A function that is a combination of two exponential functions e^x and e^-x is called hyperbolic functions. There are six hyperbolic functions like sinh, cosh, tanh, csch, sech and coth. In calculus, the derivative of hyperbolic functions can be calculated by using derivative rules.
By definition the hyperbolic differentiation is defined as:
“The process of finding derivatives of a hyperbolic function is called hyperbolic differentiation.”
The six hyperbolic functions formulas are:
$\sinh x=\frac{e^x-e^{-x}}{2}$
$\cosh x=\frac{e^x+e^{-x}}{2}$
$tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
$\text{csch x} =\frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}$
$\text{sech x}=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$
$\coth x=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$
Hyperbolic Differentiation Formulas
The derivative of a hyperbolic function can be calculated by finding the rate of change of sinh and cosh function. It is because knowing these two derivatives leads to the derivative of all other hyperbolic functions. The list of derivative formulas for hyperbolic functions is:
- $\frac{d}{dx}(\sinh x) = \cosh x$
- $\frac{d}{dx}(\cosh x) = \sinh x$
- $\frac{d}{dx}(\text{sech x}) = -\text{sech x}\tanh x$
- $\frac{d}{dx}(\text{csch x}) = \text{csch x}\coth x$
- $\frac{d}{dx}(\tanh x) = \text{sech}²x$
- $\frac{d}{dx}(\coth x) =\text{csch}²x$
The hyperbolic differentiation formula can be modified to use it with other derivative rules. Let’s discuss hyperbolic differentiation with product rule, quotient rule and power rule.
Hyperbolic Differentiation and Product Rule
If a hyperbolic function is a product of two hyperbolic functions, the product rule with the hyperbolic differentiation is used to calculate rate of change. For example, the derivative of sechx tanhx can be calculated as;
$\frac{d}{dx}(\text{sech x}\tanh x) = \text{sech x}\frac{d}{dx}[\tanh x] + \tanh x\frac{d}{dx}[\text{sech x}]$
Which is equal to,
$\frac{d}{dx}(\text{sech x}\tanh x) = \text{sech x}(\text{sech}^2 x) + \tanh x[-\text{sech x}\tanh x]=\text{sech}^3x - \tanh^2x\text{sech x}$
Since $\text{sech}^2x - \tanh^2x = 1$ then,
$\frac{d}{dx}(\text{sech x}\tanh x) = \text{sech x}(\text{sech}^2x - \tanh^2x) =\text{sech x}$
Hyperbolic Differentiation and Power Rule
Since the hyperbolic differentiation is used to calculate derivative of a hyperbolic function. It can be used along with the power rule if the function contains a hyperbolic function with power n. The relation between power rule and hyperbolic differentiation for a function $f(x) = \cosh^3x$ is expressed as;
$f’(x)=\frac{d}{dx}[\cosh^3x]$
By using power rule formula,
$f’(x,y) = 3\cosh^{3-1}|frac{d}{dx}(\cosh x)= 3\cosh^2x \sinh x$
Where, the derivative of cosh^3x with respect to x is $3\cosh^2x\sinh x$.
Hyperbolic Differentiation and Quotient Rule
If a hyperbolic function is divided by another function, the hyperbolic differentiation along with the quotient rule to find derivative. For a quotient of a function $f(x) = \tanh x = \frac{\sinh x}{\cosh x}$, the relation between hyperbolic derivative and quotient rule is,
$\frac{df}{dx}=\frac{\cosh x\frac{d}{dx}[\sinh x] – \sinh x\frac{d}{dx}[\cosh x]}{\cosh^2x}$
And,
$\frac{df}{dx}=\frac{\cosh^2x - \sinh^2x}{\cosh^2x} = \frac{1}{\cosh^2x} =\text{sech}^2x$
Hence the derivative of tanh x is $\text{sech}^2x$.
How do you do Hyperbolic differentiation step by step?
The implementation of hyperbolic derivatives is divided into a few steps. These steps assist us in calculating the derivative of a function having a hyperbolic identity. These steps are:
- Write the expression of the function.
- Identify the hyperbolic function.
- Differentiate the function with respect to the variable involved.
- Use the hyperbolic differentiation formula to calculate derivatives. For example, the derivative of sech x is -tanhx sechx.
- Simplify if needed.
Applying hyperbolic differentiation formula by using calculator
The derivative of a hyperbolic function can also be calculated by using a derivative calculator. It is an online tool that follows the hyperbolic differentiation formula to find derivatives. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of cosh x, the following steps are used by using this calculator.
- Write the expression of the function in the input box, such as cosh x.
- Choose the variable to calculate the rate of change, which will be x in this example.
- Review the input so there will be no syntax error in the function.
- Now at the last step, click on the calculate button. Using this step, the derivative calculator will provide the derivative of cosh x quickly and accurately, which will be sinh x.
Comparison between hyperbolic and trigonometric differentiation
The comparison between the hyperbolic and trigonometric differentiation can be easily analysed using the following difference table.
Hyperbolic Differentiation | Trigonometric Differentiation |
The hyperbolic differentiation is used to calculate derivative of a hyperbolic function. | The trig differentiation is used to calculate derivatives of a trigonometric function. |
There are different formulas to calculate derivative of hyperbolic functions. | There are different formulas to calculate derivative of trigonometric functions. |
The hyperbolic differentiation can be used along with different derivative formulas. | The derivative of all trigonometric functions can be calculated by using product rule, quotient rule and power rule. |
Conclusion
The hyperbolic functions combine two exponential functions, e^{x} and e^{-x}. Calculating the derivative of such functions is known as hyperbolic function differentiation. In conclusion, the hyperbolic differentiation of all trig functions can be computed using the sinh and cosh derivatives.