How to Derive the Derivative of Sec x Using the Quotient Rule and the Fact that Sec x = 1/Cos x

Derivative of sec x

To find the derivative of sec x, we’ll use the quotient rule and the fact that sec x can be expressed as 1/cos x

To find the derivative of sec x, we’ll use the quotient rule and the fact that sec x can be expressed as 1/cos x.

Let f(x) = sec x = 1/cos x.

Using the quotient rule, we have:

f'(x) = (d/dx)(1/cos x)
= (0*cos x – 1*(-sin x))/(cos^2 x) [Applying quotient rule]
= -(-sin x)/(cos^2 x)
= sin x/(cos^2 x)
= sin x * (1/cos^2 x)
= sin x * sec^2 x

Therefore, the derivative of sec x is sin x times sec^2 x, or f'(x) = sin x * sec^2 x.

Note: The derivative of sec x can also be derived using the chain rule by considering sec x as (cos x)^(-1) and then differentiating it. However, using the quotient rule is a more direct approach.

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