Derivative of Inverse Trigonometric Functions 

Inverse trigonometric functions play a crucial role in calculus and are widely used in differentiation problems. The derivatives of inverse trigonometric functions are essential for solving integrals, limits, and various mathematical applications. These functions help to reverse the effects of standard trigonometric functions. Understanding their derivatives, along with their formulas and applications, is vital for students, especially in competitive exams like JEE and board exams. This guide covers formulas, solved examples, and important practice questions. 

1.0What Are Inverse Trigonometric Functions?

Inverse trigonometric functions are the inverses of standard trigonometric functions. They are used to find the angle when the value of the trigonometric function is known.

The most common inverse trigonometric functions include:

• $Si{n}^{-1}x(arctan)$
• $Co{s}^{-1}x(arccos)$
• $ta{n}^{-1}x(arctan)$
• $co{t}^{-1}x(arccot)$
• $Se{c}^{-1}x(arcsec)$
• $Cose{c}^{-1}x(arccosec)$

2.0Derivatives of Inverse Trigonometric Functions Formulas

Here are the standard formulas for derivatives of inverse trigonometric functions:

3.0Derivative Inverse Trigonometric Functions: General Rule

For a composite function of the form y = f(g(x)),

The derivative is: $\frac{dy}{dx}=f^{\prime }(g(x)).g^{\prime }(x)$

Example: $y=si{n}^{-1}(2x)$

Derivative: $\frac{dy}{dx}=\frac{1}{\sqrt{1-2(x{)}^2}}.2=\frac{2}{\sqrt{1-4x^2}}$

4.0Solved Examples on Derivatives of Inverse Trigonometric Functions

Example 1: Differentiate $y=si{n}^{-1}(\frac{x}{2})$

Solution:

$\frac{dy}{dx}=\frac{1}{\sqrt{1-(\frac{x}{2}{)}^2}}.\frac{1}{2}$

$\frac{dy}{dx}=\frac{1}{2\sqrt{1-\frac{x^2}{4}}}$

$\frac{dy}{dx}=\frac{1}{\sqrt{4-x^2}}$

Example 2: Find $\frac{d}{dx}(ta{n}^{-1}\sqrt{x})$.

Solution:

$\frac{d}{dx}\left({\tan }^{-1}\sqrt{x}\right)=\frac{1}{1+(\sqrt{x}{)}^2}\cdot \frac{1}{2\sqrt{x}}=\frac{1}{1+x}\cdot \frac{1}{2\sqrt{x}}=\frac{1}{2\sqrt{x}(1+x)}$

Example 3: Differentiate $y=se{c}^{-1}(3x)$.

Solution:

$\frac{dy}{dx}=\frac{1}{|3x|\sqrt{(3x{)}^2-1}}$

$\frac{dy}{dx}=\frac{3}{|3x|\sqrt{9x^2-1}}$

$\frac{dy}{dx}=\frac{1}{|x|\sqrt{9x^2-1}}$

Example 4: Find $\frac{d}{dx}\left({\sin }^{-1}\frac{x}{2}\right)$.

Solution:

We apply chain rule: $\frac{d}{dx}\left({\sin }^{-1}u\right)=\frac{1}{\sqrt{1-u^2}}\cdot \frac{du}{dx}$

Here, $u=\frac{x}{2}$:

$\frac{dy}{dx}=\frac{1}{\sqrt{1-{\left(\frac{x}{2}\right)}^2}}\cdot \frac{1}{2}$

$\frac{dy}{dx}=\frac{1}{2\sqrt{1-\frac{x^2}{4}}}=\frac{1}{\sqrt{4-x^2}}$

Example 5: Differentiate $y=ta{n}^{-1}\sqrt{x}$.

Solution:

$\frac{dy}{dx}=\frac{1}{1+(\sqrt{x}{)}^2}\cdot \frac{1}{2\sqrt{x}}$

$\frac{dy}{dx}=\frac{1}{1+x}\cdot \frac{1}{2\sqrt{x}}$

$\frac{dy}{dx}=\frac{1}{2\sqrt{x}(1+x)}$

Example 6: Find $\frac{d}{dx}(se{c}^{-1}(3x))$.

Solution:

$\frac{dy}{dx}=\frac{1}{|3x|\sqrt{(3x{)}^2-1}}.3$

$\frac{dy}{dx}=\frac{3}{|3x|\sqrt{9x^2-1}}$

$\frac{dy}{dx}=\frac{1}{|x|\sqrt{9x^2-1}}$

Example 7: Find derivative of $y=co{s}^{-1}(2x)$.

Solution:

$\frac{dy}{dx}=-\frac{1}{\sqrt{1-(2x{)}^2}}.2$

$\frac{dy}{dx}=-\frac{2}{\sqrt{1-4x^2}}$

5.0Practice Questions on Derivatives of Inverse Trigonometric Functions

1. Differentiate $y=si{n}^{-1}(\frac{3x}{4})$.
2. Find $\frac{d}{dx}(co{t}^{-1}\sqrt{1-x^2})$.
3. Differentiate $y=cs{c}^{-1}(\frac{x}{2})$.
4. Compute $\frac{d}{dx}(ta{n}^{-1}(\frac{x}{1-x}))$.
5. Find $\frac{dy}{dx}$ if $y=se{c}^{-1}(\frac{1}{x})$.

6.0Tips to Remember Derivatives of Inverse Trigonometric Functions:

1. Always check domain restrictions before differentiating.
2. Watch out for chain rule when function is composite.
3. For $se{c}^{-1}x$ and $cs{c}^{-1}x$, always use absolute value in derivative.
4. Common JEE trick: Rationalize square roots if needed.

7.0Applications of Derivatives of Inverse Trigonometric Functions

• Solving integrals involving inverse trig functions.
• Solving differential equations.
• Coordinate geometry problems.
• Limits involving inverse trigonometric expressions.
• Physics problems involving angular relations.

8.0Sample Questions on Derivatives of Inverse Trigonometric Functions

Q1. What is the derivative of inverse sine function?

Ans:  $\frac{d}{dx}(si{n}^{-1}x)=\frac{1}{\sqrt{1-x^2}}$

Q2. What is the derivative of inverse tangent function?

Ans:  $\frac{d}{dx}(ta{n}^{-1}x)=\frac{1}{1+x^2}$

Q3. What are the common mistakes in differentiation of inverse trig functions?

Ans: 

• Missing chain rule
• Ignoring absolute values in $se{c}^{-1}x$ and $cs{c}^{-1}x$
• Misapplying square roots