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)

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