Here you will learn what is the differentiation of inverse trigonometric functions with examples.
Let’s begin –
Differentiation of Inverse Trigonometric Functions
(i) If x \(\in\) (-1, 1), then the differentiation of \(sin^{-1}x\) or arcsinx with respect to x is \(1\over \sqrt{1-x^2}\).
i.e. \(d\over dx\) \(sin^{-1}x\) = \(1\over \sqrt{1-x^2}\) , for x \(\in\) (-1, 1).
(ii) If x \(\in\) (-1, 1), then the differentiation of \(cos^{-1}x\) or arccosx with respect to x is \(-1\over \sqrt{1-x^2}\).
i.e. \(d\over dx\) \(cos^{-1}x\) = \(-1\over \sqrt{1-x^2}\) , for x \(\in\) (-1, 1).
(iii) The differentiation of \(tan^{-1}x\) or arctanx with respect to x is \(1\over {1+x^2}\).
i.e. \(d\over dx\) \(tan^{-1}x\) = \(1\over {1+x^2}\).
(iv) The differentiation of \(cot^{-1}x\) or arccotx with respect to x is \(-1\over {1+x^2}\).
i.e. \(d\over dx\) \(cot^{-1}x\) = \(-1\over {1+x^2}\).
(v) If x \(\in\) R – [-1, 1], then the differentiation of \(sec^{-1}x\) or arcsecx with respect to x is \(1\over |x|\sqrt{x^2-1}\).
i.e. \(d\over dx\) \(sec^{-1}x\) = \(1\over |x|\sqrt{x^2-1}\) , x \(\in\) R – [-1, 1]
(vi) If x \(\in\) R – [-1, 1], then the differentiation of \(cosec^{-1}x\) or arccosecx with respect to x is \(-1\over |x|\sqrt{x^2-1}\).
i.e. \(d\over dx\) \(cosec^{-1}x\) = \(-1\over |x|\sqrt{x^2-1}\) , x \(\in\) R – [-1, 1]
Example 1 : find the differentiation of \(sin^{-1}5x\).
Solution : Let y = \(sin^{-1}5x\)
Now, \(dy\over dx\) = \(1\over \sqrt{1-(5x)^2}\).5
= \(5\over \sqrt{1-25x^2}\)
Example 2 : find the differentiation of \(cos^{-1}5x\).
Solution : Let y = \(cos^{-1}5x\)
Now, \(dy\over dx\) = \(-1\over \sqrt{1-(5x)^2}\).5
= \(-5\over \sqrt{1-25x^2}\)
Example 3 : find the differentiation of \(tan^{-1}5x\).
Solution : Let y = \(tan^{-1}5x\)
Now, \(dy\over dx\) = \(1\over {1+(5x)^2}\).5
= \(5\over {1+25x^2}\)
Example 4 : find the differentiation of \(sec^{-1}5x\).
Solution : Let y = \(sec^{-1}5x\)
Now, \(dy\over dx\) = \(1\over |5x|\sqrt{(5x)^2 – 1}\).5
= \(5\over |5x|\sqrt{25x^2-1}\) = \(1\over x\sqrt{25x^2-1}\)
Example 5 : find the differentiation of \(cosec^{-1}5x\).
Solution : Let y = \(cosec^{-1}5x\)
Now, \(dy\over dx\) = \(-1\over |5x|\sqrt{1-(5x)^2}\).5
= \(-5\over |5x|\sqrt{25x^2-1}\) = \(-1\over x\sqrt{25x^2-1}\)