Proof of Derivative of sec2x

There are two methods to find derivative of sec²x

• Using Chain Rule of Differentiation
• Using Quotient Rule
• Using First Principles of Derivatives

Derivative of sec²x using Chain Rule of Differentiation

Chain Rule of differentiation states that for a composite function f(g(x)),

[f{g(x)}]’ = f'{g(x)} × g'(x)

Therefore applying chain rule to f(x) = sec²x, we get,

⇒ f'(x) = 2secx × (secx)’

⇒ f'(x) = 2secx × (secx.tanx)

⇒ f'(x) = 2sec²x.tanx

Thus, we have derived the derivative of f(x) = sec²x using the chain rule.

Derivative of sec²x Using Quotient Rule

Quotient rule in differentiation states that,

For two functions u and v the differentiation of (u/v) is found as,

(u/v)’ = (vu’ – uv’)/v²

Now f(x) = sec²x can be written as f(x) = 1/cos²x

Applying quotient rule for f(x) = 1/cos²x, we get,

⇒ f'(x) = (cos²x(1)’ – (1)(cos²x)’)/(cos⁴x)

Now, we know that, (cosx)’ = -sinx

⇒ f'(x) = [-2cosx.(-sinx)]/(cos⁴x)

On simplification, we get

⇒ f'(x) = 2sec²x.tanx

Thus, we obtain the same result for derivative of sec²x by quotient rule.

Derivative of sec²x using First Principle of Derivatives

First principle of differentiation state that derivative of a function f(x) is defined as,

f'(x) = lim_h→0 [f(x + h) – f(x)]/[(x + h) – x]

This can also be represented as,

f'(x) = lim_h→0 [f(x + h) – f(x)]/ h

Putting f(x) = sec²x, to find derivative of sec²x, we get,

⇒ f'(x) = lim_h→0 [sec²(x + h) – sec²x]/ h

⇒ f'(x) = lim_h→0 (sec(x+h) + sec(x)).(sec(x+h) – sec(x))/h

⇒ f'(x) = lim_h→0 (sec(x+h) + sec(x)).(1/cos(x+h) – 1/cos(x))/h

⇒ f'(x) = lim_h→0 (sec(x+h) + sec(x)).(cos(x) – cos(x+h))/hcos(x+h)cos(x)

Using, cos(A + B) = cosAcosB – sinAsinB, we get,

⇒ f'(x) = lim_h→0 (sec(x+h) + sec(x)).(cosx – cosxcosh + sinxsinh)/hcos(x+h)cos(x)

⇒ f'(x) = lim_h→0 (sec(x+h) + sec(x)).(cosx(1 – cosh) + sinxsinh)/hcos(x+h)cos(x)

Now, putting lim_h→0(1-cosh)/h = 0 and lim_h→0(sinh)/h = 1, we get,

⇒ f'(x) = lim_h→0(sec(x+h) + sec(x)).(sinx)/cos(x+h)cosx

⇒ f'(x) = (sec(x+0) + sec(x)).(sinx)/cos(x+0)cosx

⇒ f'(x) = (2secxsinx)/cos²x

⇒ f'(x) = 2sec²xtanx

Thus, derivative of sec²x has been derived using first principle of differentiation.

Read More,

• Derivative in Math
• Derivative of Secant x
• Derivative of Trigonometric Function

Derivative of sec²x is 2sec²xtanx. Sec²x is the square of the trigonometric function secant x, generally written as sec x.

In this article, we will discuss the derivative of sec^2x, various methods to find it including the chain rule and the quotient rule, solved examples, and some practice problems on it.