Introduction to Inverse hyperbolic Differentiation
There are six basic inverse hyperbolic functions in geometry. The rate of change of these functions is known as inverse hyperbolic differentiation. This method follows the derivative rules and formulas to calculate rate of change. Let’s understand how to apply inverse trigonometric differentiation step-by-step and also learn the difference between trig and inverse differentiation.
Understanding the Inverse Hyperbolic Differentiation
A function that is the inverse relation of the combination of two exponential functions e^x and e^-x is called inverse hyperbolic function. There are six basic inverse hyperbolic functions like arsinh, arcosh, artanh, arcsch, arsech and arcoth. In calculus, the derivative of inverse hyperbolic functions can be calculated by using derivative rules.
By definition the inverse hyperbolic differentiation is defined as:
“The process of finding derivative of an inverse hyperbolic function is called inverse hyperbolic differentiation.”
Inverse Hyperbolic Differentiation Formulas
The derivative of an inverse hyperbolic function can be calculated by finding the rate of change of arcsinh and arcosh function. It is because knowing these two derivatives leads to the derivative of all other inverse hyperbolic functions. The list of derivative formulas for inverse hyperbolic functions is:
- $\frac{d}{dx}(\sinh^{-1}x) = \frac{1}{\sqrt{1+x^2}}$
- $\frac{d}{dx}(\cosh^{-1}x) = \frac{1}{\sqrt{x^2-1}}$
- $\frac{d}{dx}(\tanh^{-1}x) = \frac{1}{1-x^2}$
- $\frac{d}{dx}(\text{csch}^{-1}x) = -\frac{1}{|x|\sqrt{1+x^2}}$
- $\frac{d}{dx}(\text{sech}^{-1}x) = -\frac{1}{x\sqrt{1-x^2}}$
- $\frac{d}{dx}(\text{coth}^{-1}x) = \frac{1}{1-x^2}$
The inverse hyperbolic trig differentiation formula can be modified to use it with other derivative rules. Let’s discuss the inverse hyperbolic differentiation with product rule, quotient rule and power rule.
Inverse Hyperbolic Differentiation and Product Rule
If an inverse hyper function is a product of two functions, the product rule with the inverse hyperbolic differentiation is used to calculate rate of change. The product rule formula can be written as;
$\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)]$
Which can be also written as,
$\frac{d}{dx}(f(x)g(x)) = f(x)g’(x) + g(x)f’(x)$
Inverse hyperbolic Differentiation and Power Rule
Since the inverse hyperbolic differentiation is used to calculate derivative of an inverse hyperbolic function. It can be used along with the power rule if the function contains an inverse hyperbolic function with power n. The relation between power rule and inverse hyperbolic differentiation for a function f(x) = (tanh-1x)2 is expressed as;
$f’(x) = \frac{d}{dx}[\tanh^{-1}x]^2$
By using power rule formula,
$f’(x) = 2\tanh^{-1}x\frac{d}{dx}(\tanh^{-1}x)= 2\tan^{-1}x\left(\frac{1}{1-x^2}\right)$
Where, the tanh^{-1}x derivative with respect to x is $\frac{1}{1-x^2}$.
Inverse Hyperbolic Differentiation and Quotient Rule
If an inverse hyperbolic function is divided by another function, the inverse hyperbolic differentiation along with the quotient rule to find derivative. For a quotient of a function f(x)/g(x) , the relation between inverse hyperbolic derivative and quotient rule is,
$\frac{df}{dx} =\frac{g(x)\frac{d}{dx}[f(x)] – f(x)\frac{d}{dx}[g(x)]}{(g(x))^2}$
How do you do Inverse Trigonometric differentiation step by step?
The implementation of derivative of inverse hyperbolic functions is divided into a few steps. These steps assist us to calculate the derivative of a function having a trigonometric identity. These steps are:
- Write the expression of the function.
- Identify the inverse hyperbolic function.
- Differentiate the function with respect to the variable involved.
- Use the inverse hyperbolic function differentiation formula to calculate derivatives. For example the derivative of arccosh is $\frac{-1}{1-x^2}$.
- Simplify if needed.
Differentiation formula on inverse hyperbolic functions by using calculator
The derivative of an inverse hyperbolic function can be also calculated by using derivative calculator. It is an online tool that follows the differentiation formula of inverse hyperbolic functions to find derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of arsinh, the following steps are used by using this calculator.
- Write the expression of the function in the input box such as, arcosh x.
- Choose the variable to calculate the rate of change, which will be x in this example.
- Review the input so that there will be no syntax error in the function.
- Now at the last step, click on the calculate button. By using this step, the derivative calculator will provide the derivative of arcos quickly and accurately which will be 1/√x²+1.
Comparison between hyperbolic and inverse hyperbolic differentiation
The comparison between the hyperbolic and inverse hyperbolic differentiation can be easily analysed using the following difference table.
Inverse Hyperbolic Differentiation | Hyperbolic Differentiation |
The differentiation of inverse hyperbolic formula is used to calculate the derivative of an inverse hyperbolic function. | The hyperbolic differentiation is used to calculate derivative of a hyperbolic function. |
There are difference formulas to calculate derivative of hyperbolic inverse functions. | There are different formulas to calculate derivative of hyperbolic function. |
The inverse hyperbolic differentiation can be used along with different derivative formulas. | The derivative of all hyperbolic functions can be calculated by using product rule, quotient rule and power rule. |
Conclusion
The inverse of hyperbolic differentiation is a method to find rate of change in inverse hyperbolic functions. This method uses all derivative rules to calculate derivatives. In conclusion, we can say the derivative of all inverse hyperbolic functions can be calculated by using the derivative of arsinh and arcosh.