Derivative of sec(x): A Breakdown and Alternative Representation

d/dx(secx)

secxtanx

To find the derivative of sec(x) with respect to x, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = u(x)/v(x), where u(x) and v(x) are both functions of x, then the derivative of f(x) with respect to x is given by:

f'(x) = (v(x) * u'(x) – u(x) * v'(x))/[v(x)]^2

Applying this rule to the function f(x) = sec(x), we have:

u(x) = 1 (since sec(x) can be written as 1/cos(x))
u'(x) = 0 (the derivative of a constant is 0)

v(x) = cos(x)
v'(x) = -sin(x) (the derivative of cos(x) is -sin(x))

Using these values, we can compute the derivative of sec(x):

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

Therefore, the derivative of sec(x) with respect to x is sin(x)/[cos(x)]^2.

This result can also be written in terms of the secant function itself. Recall that sec^2(x) = 1 + tan^2(x), so we can substitute this into the derivative expression:

f'(x) = sin(x)/[cos(x)]^2
= sin(x)/(1/cos^2(x))
= sin(x)cos^2(x)
= sin(x)sec^2(x)

Hence, we have found an alternative representation of the derivative of sec(x) as sin(x)sec^2(x).

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