Differentiation of Inverse Trigonometric Functions

In this tutorial we shall discuss the basic formulas of differentiation for inverse trigonometric functions.

 

1. \frac{d}{{dx}}{\sin ^{ - 1}}x = \frac{1}{{\sqrt {1 - {x^2}} }}

2. \frac{d}{{dx}}{\cos ^{ - 1}}x = - \frac{1}{{\sqrt {1 - {x^2}} }}

3. \frac{d}{{dx}}{\tan ^{ - 1}}x = \frac{1}{{1 + {x^2}}}

4. \frac{d}{{dx}}{\cot ^{ - 1}}x = - \frac{1}{{1 + {x^2}}}

5. \frac{d}{{dx}}{\sec ^{ - 1}}x = \frac{1}{{x\sqrt {{x^2} - 1} }}

6. \frac{d}{{dx}}{\csc ^{ - 1}}x = - \frac{1}{{x\sqrt {{x^2} - 1} }}

 

To remember these formulas, one point to be noted is that these functions come with negative signs starting with the letter C.

 

These are the general inverse trigonometric formulas for functions with angles:

 

1. \frac{d}{{dx}}{\sin ^{ - 1}}f\left( x \right) = \frac{1}{{\sqrt {1 - {{\left[ {f\left( x \right)} \right]}^2}} }}\frac{d}{{dx}}f\left( x \right)

2. \frac{d}{{dx}}{\cos ^{ - 1}}f\left( x \right) = - \frac{1}{{\sqrt {1 - {{\left[ {f\left( x \right)} \right]}^2}} }}\frac{d}{{dx}}f\left( x \right)

3. \frac{d}{{dx}}{\tan ^{ - 1}}f\left( x \right) = \frac{1}{{1 + {{\left[ {f\left( x \right)} \right]}^2}}}\frac{d}{{dx}}f\left( x \right)

4. \frac{d}{{dx}}{\cot ^{ - 1}}f\left( x \right) = - \frac{1}{{1 + {{\left[ {f\left( x \right)} \right]}^2}}}\frac{d}{{dx}}f\left( x \right)

5. \frac{d}{{dx}}{\sec ^{ - 1}}f\left( x \right) = \frac{1}{{f\left( x \right)\sqrt {{{\left[ {f\left( x \right)} \right]}^2} - 1} }}\frac{d}{{dx}}f\left( x \right)

6. \frac{d}{{dx}}{\csc ^{ - 1}}f\left( x \right) = - \frac{1}{{f\left( x \right)\sqrt {{{\left[ {f\left( x \right)} \right]}^2} - 1} }}\frac{d}{{dx}}f\left( x \right)