Here you will learn differentiation of cot inverse x or arccotx x by using chain rule.<\/p>\n
Let’s begin –<\/p>\n
\nThe differentiation of \\(cot^{-1}x\\) with respect to x is \\(-1\\over {1 + x^2}\\).<\/p>\n
i.e. \\(d\\over dx\\) \\(cot^{-1}x\\) = \\(-1\\over {1 + x^2}\\).<\/p>\n<\/blockquote>\n
Proof using chain rule :<\/h2>\n
\nLet y = \\(cot^{-1}x\\). Then,<\/p>\n
\\(cot(cot^{-1}x)\\) = x<\/p>\n
\\(\\implies\\) cot y = x<\/p>\n
Differentiating both sides with respect to x, we get<\/p>\n
\\(d\\over dx\\)(cot y) = \\(d\\over dx\\)(x)<\/p>\n
\\(d\\over dx\\) (cot y) = 1<\/p>\n
By chain rule,<\/p>\n
\\(-cosec^2 y\\) \\(dy\\over dx\\) = 1<\/p>\n
\\(dy\\over dx\\) = \\(-1\\over cosec^2 y\\)<\/p>\n
[ \\(\\because\\) 1 + \\(cot^2 y\\) = \\(cosec^2 y\\)<\/p>\n
\\(dy\\over dx\\) = \\(1\\over {1 + cot^2 y}\\)<\/p>\n
\\(\\implies\\) \\(dy\\over dx\\) = \\(-1\\over {1 + x^2}\\)<\/p>\n
\\(\\implies\\) \\(d\\over dx\\) \\(cot^{-1}x\\) = \\(-1\\over {1 + x^2}\\)\u00a0<\/p>\n
Hence, the differentiation of \\(cot^{-1}x\\) with respect to x is \\(-1\\over {1 + x^2}\\).<\/p>\n<\/blockquote>\n
Example<\/strong><\/span> : What is the differentiation of \\(cot^{-1} x^2\\) with respect to x ?<\/p>\n
Solution<\/strong><\/span> : Let y = \\(cot^{-1} x^2\\)<\/p>\n
Differentiating both sides with respect to x and using chain rule, we get<\/p>\n
\\(dy\\over dx\\) = \\(d\\over dx\\) (\\(cot^{-1} x^2\\))<\/p>\n
\\(dy\\over dx\\) = \\(-1\\over {1 + x^4}\\).(2x) = \\(-2x\\over {1 + x^4}\\)<\/p>\n
Hence, \\(d\\over dx\\) (\\(cot^{-1} x^2\\)) = \\(-2x\\over {1 + x^4}\\)<\/p>\n
Example<\/strong><\/span> : What is the differentiation of x + \\(cot^{-1} x\\) with respect to x ?<\/p>\n
Solution<\/strong><\/span> : Let y = x + \\(cot^{-1} x\\)<\/p>\n
Differentiating both sides with respect to x, we get<\/p>\n
\\(dy\\over dx\\) = \\(d\\over dx\\) (x) + \\(d\\over dx\\) (\\(cot^{-1} x\\))<\/p>\n
\\(dy\\over dx\\) = 1 + \\(-1\\over {1 + x^2}\\)<\/p>\n
Hence, \\(d\\over dx\\) (x + \\(cot^{-1} x\\)) = 1 – \\(1\\over {1 + x^2}\\)<\/p>\n
\nRelated Questions<\/h3>\n
What is the Differentiation of cot x ?<\/a><\/p>\n
What is the Integration of Cot Inverse x ?<\/a><\/p>\n
What is the Differentiation of tan inverse x ?<\/a><\/p>\n\n\n