Differentiation of Inverse Trigonometric Functions

In this tutorial we discuss 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} }}

For remember these formulas one point to be noted that those function comes with negative sign starting with alphabet C.

Now general inverse trigonometric formulae when any function is given in terms angles, the following formula of the form

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)