Examples on Derivative of sec2x

Various examples on derivative of sec²x

Example 1: Find the derivative of f(x) = sec²(x²+9)

Solution:

We have, f(x) = sec²(x²+9)

By applying chain rule,

⇒ f'(x) = 2sec²(x²+9)×tan(x²+9)×(x²+9)’

⇒ f'(x) = 2sec²(x²+9)×tan(x²+9)×(2x)

⇒ f'(x) = 4x.sec²(x²+9).tan(x²+9)

Example 2: Find the derivative of f(x) = x.sec²x

Solution:

We have, f(x) = xsec²x

By applying product rule,

⇒ f'(x) = x(sec²x)’ + (x)’sec²x

⇒ f'(x) = x.2sec²xtanx + sec²x

⇒ f'(x) = sec²x(2xtanx + 1)

Example 3: Find the derivative of f(x) = x/sec²x

Solution:

We have, f(x) = x/sec²x

By applying product rule,

⇒ f'(x) = [(sec²x)(x)’ – (x)(sec²x)’]/[sec²x]²

⇒ f'(x) = [sec²x – x(2sec²xtanx)]/sec⁴x

⇒ f'(x) = [(sec²x)(1-2xtanx)]/sec⁴x

⇒ f'(x) = (1-2xtanx)/sec²x

⇒ f'(x) = cos²x – 2x.sinx.cosx

⇒ f'(x) = cos²x – xsin2x

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.