d/dx [secx]
secxtanx
To differentiate sec(x) with respect to x, we can use the chain rule and the fact that the derivative of cos(x) is -sin(x).
Let u(x) = cos(x), so sec(x) = 1/cos(x) = 1/u(x).
Then, by the chain rule:
d/dx [sec(x)] = d/dx [1/u(x)] = (-1/u(x)^2) * u'(x)
We can simplify this using the trigonometric identity for the derivative of cosine:
u'(x) = -sin(x)
Substituting this into our expression:
d/dx [sec(x)] = (-1/cos(x)^2) * (-sin(x))
= sin(x)/cos(x)^2
= sin(x)sec(x)^2
Therefore, the derivative of sec(x) with respect to x is sin(x)sec(x)^2.
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