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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