d/dx(sec(x))
To find the derivative of sec(x) with respect to x, we can use the quotient rule
To find the derivative of sec(x) with respect to x, we can use the quotient rule. The derivative of sec(x) can be written as:
d/dx(sec(x)) = d/dx(1/cos(x))
Applying the quotient rule, we have:
= (cos(x)(0) – 1(sin(x)))/(cos(x))^2
= -sin(x)/(cos(x))^2
Using trigonometric identities, we can simplify further. We know that sec(x) = 1/cos(x) and cos^2(x) = 1 – sin^2(x). So, substituting these identities into our derivative, we get:
= -sin(x)/(1 – sin^2(x))
= -sin(x)/cos^2(x)
= -tan(x)/cos(x)
= -tan(x)sec(x)
Therefore, the derivative of sec(x) with respect to x is -tan(x)sec(x).
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