Here you will learn what is the differentiation of secx and its proof by using first principle.
Let’s begin –
Differentiation of secx
The differentiation of secx with respect to x is secx.tanx
i.e. \(d\over dx\) (secx) = secx.tanx
Proof Using First Principle :
Let f(x) = sec x. Then, f(x + h) = sec(x + h)
\(\therefore\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(f(x + h) – f(x)\over h\)
\(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(sec(x + h) – sec x\over h\)
\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \({1\over cos(x + h)} – {1\over cos x}\over h\)
\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(cos x – cos(x + h)\over h cos x cos(x +h)\)
By using trigonometry formula,
[cos C – cos D = \(2sin ({C + D\over 2})sin ({D – C\over 2})\)]
\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(2sin ({x + x + h\over 2})sin({x + h – x\over 2})\over h cos x cos (x + h)\)
\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(2sin ({2x + h\over 2})sin({h\over 2})\over h cos x cos (x + h)\)
\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(sin ({2x + h\over 2})\over cos x cos(x + h)\).\(lim_{h\to 0}\) \(sin(h/2)\over (h/2)\)
because, [\(lim_{h\to 0}\)\(sin(h/2)\over (h/2)\) = 1]
\(\implies\) \(d\over dx\)(f(x)) = \(sin x\over cos x cos x\)(1) = tan x sec x
Hence, \(d\over dx\) (sec x) = secx.tanx
Example : What is the differentiation of sec x + x with respect to x?
Solution : Let y = sec x + x
\(d\over dx\)(y) = \(d\over dx\)(sec x + x)
\(\implies\) \(d\over dx\)(y) = \(d\over dx\)(sec x) + \(d\over dx\)(x)
By using secx differentiation we get,
\(\implies\) \(d\over dx\)(y) = sec x tan x + 1
Hence, \(d\over dx\)(sec x + x) = sec x tan x + 1
Example : What is the differentiation of \(sec\sqrt{x}\) with respect to x?
Solution : Let y = \(sec\sqrt{x}\)
\(d\over dx\)(y) = \(d\over dx\)(\(sec\sqrt{x}\))
By using chain rule we get,
\(\implies\) \(d\over dx\)(y) = \(1\over 2\sqrt{x}\)(\(sec \sqrt{x}.tan\sqrt{x}\))
Hence, \(d\over dx\)(\(sec\sqrt{x}\)) = \(1\over 2\sqrt{x}\)(\(sec\sqrt{x}.tan\sqrt{x}\))
Related Questions
What is the Differentiation of sec inverse x ?
