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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