Introduction to Inverse Trigonometric differentiation
There are six basic inverse trigonometric functions in geometry. The rate of change of these functions is known as inverse trigonometric differentiation. This method follows the derivative rules and formulas to calculate the rate of change. Let’s understand how to apply inverse trigonometric differentiation step-by-step and learn the difference between trig and inverse differentiation.
Understanding the Inverse Trigonometric Differentiation
A function that is the inverse relation of an angle to a right-angled triangle is known as an inverse trigonometric function. There are six basic inverse trigonometric functions arsine, arcos, artan, arcsc, arsec, and arcot. In calculus, the derivative of inverse trigonometric functions can be calculated using derivative rules.
By definition, the inverse trigonometric differentiation is defined as:
“The process of finding the derivative of an inverse trig function is called inverse trig differentiation.”
Inverse Trigonometric Differentiation Formulas
The derivative of an inverse trigonometric function can be calculated by finding the rate of change of the arcsine functions. It is because knowing these two derivatives leads to the derivative of all other inverse trig functions. The list of derivative formulas for inverse trigonometric functions is as follows:
- $\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}$
- $\frac{d}{dx}(\csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}}$
- $\frac{d}{dx}(\sec^{-1}x)= \frac{1}{|x|\sqrt{x^-1}}$
- $\frac{d}{dx}(\cot^{-1}x)=\frac{-1}{1+x^2}$
The inverse trigonometric differentiation formula can be modified to use it with other derivative rules. Let’s discuss the inverse trig differentiation with the product rule, quotient rule, and power rule.
Inverse Trigonometric Differentiation and Product Rule
If an inverse trig function is a product of two functions, the product rule with the inverse trigonometric differentiation is used to calculate the rate of change. The product rule formula can be written as follows;
$$\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)]$$
Which can be also written as,
$$\frac{d}{dx}(f(x)g(x)) = f(x)g’(x) + g(x)f’(x)$$
Inverse Trigonometric Differentiation and Power Rule
Since the inverse trigonometric differentiation is used to calculate derivative of an inverse trigonometric function. It can be used along with the power rule if the function contains an inverse trig function with power n. The relation between power rule and trig differentiation for a function f(x) = (tan-1x)2 is expressed as;
$$f’(x) = \frac{d}{dx}[\tan^{-1}x]^2$$
By using power rule formula,
$$f’(x) = 2\tan^{-1}x\frac{d}{dx}(\tan^{-1}x)= 2\tan^{-1}x\left(\frac{1}{1+x^2}\right)$$
Where, the derivative of tan with respect to x is $\frac{1}{1+x^2}$.
Inverse Trigonometric Differentiation and Quotient Rule
If an inverse trigonometric function is divided by another function, the inverse trig differentiation, along with the quotient rule to find the derivative. For a quotient of a function f(x)/g(x), the relation between inverse trig 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 the inverse trigonometric derivative is divided into a few steps. These steps assist us in calculating the derivative of a function having a trigonometric identity. These steps are:
- Write the expression of the function.
- Identify the inverse trig function.
- Differentiate the function with respect to the variable involved.
- Use the inverse trigonometric differentiation formula to calculate derivatives. For example, the derivative of arcos is -1/√1-x².
- Simplify if needed.
Applying inverse trigonometric differentiation formula by using calculator
The derivative of an inverse log function can also be calculated using the derivative calculator. It is an online tool that follows the inverse trig differentiation formula to find the derivative. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of arcos, the following steps are used by using this calculator.
- Write the expression of the function in the input box, such as arcos 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 arcos quickly and accurately, which will be -1/√1-x².
Comparison between Trigonometric and inverse trigonometric differentiation
The comparison between the trigonometric and inverse trigonometric differentiation can be easily analysed using the following difference table.
Inverse Trigonometric Differentiation |
Trigonometric Differentiation |
The inverse trig differentiation is used to calculate the derivative of an inverse trigonometric function. |
The trig differentiation is used to calculate derivative of a trigonometric function. |
There are difference formulas to calculate derivative of inverse trig functions. |
There are different formulas to calculate derivative of trigonometric function. |
The inverse trig 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 inverse trigonometric differentiation is a method to find rate of change in inverse trig functions. This method uses all derivative rules to calculate derivatives. In conclusion, we can say the derivative of all inverse trigonometric functions can be calculated by using the derivative of arsin and arcos.