# Differentiation of secx

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 ?

What is the Differentiation of cosx ?

What is the Integration of Secx ?