How To Compute The Derivative Of Sec(X) Using Quotient Rule And Reciprocal Of Cos(X)

derv sec(x)

(sec(x)tan(x))

The derivative of sec(x) can be found using the quotient rule and the fact that sec(x) is the reciprocal of cos(x).

Let’s start by expressing sec(x) as 1/cos(x). Then we can apply the quotient rule:

[f(x) / g(x)]’ = [f'(x)*g(x) – g'(x)*f(x)] / [g(x)^2]

Where f(x) = 1 and g(x) = cos(x), we have:

[1/cos(x)]’ = [(0*cos(x) – (-sin(x)*1)) / (cos(x))^2]

Simplifying, we get:

[1/cos(x)]’ = sin(x) / cos(x)^2

Using the identity sin(x) = 1/csc(x) and replacing cos(x)^2 with 1/sec(x)^2, the final answer is:

[1/cos(x)]’ = sin(x)/cos(x)^2 = 1/(cos(x) * sin(x)) = csc(x) * sec(x)

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