Derivative of Sec(x) with Respect to x: Quotient Rule Combined with Chain Rule

d/dx(sec(x))

To find the derivative of sec(x) with respect to x, we can use the quotient rule combined with the chain rule

To find the derivative of sec(x) with respect to x, we can use the quotient rule combined with the chain rule.

The quotient rule states that for two functions u(x) and v(x), (u/v)’ = (u’v – uv’) / v^2.

In this case, u(x) = 1 and v(x) = cos(x). Therefore, u'(x) = 0 and v'(x) = -sin(x).

Differentiating sec(x) with respect to x, we have:

d/dx(sec(x)) = (d/dx(1) * cos(x) – 1 * -sin(x)) / cos^2(x)
= (0 * cos(x) – sin(x)) / cos^2(x)
= -sin(x) / cos^2(x)
= -sin(x)/cos(x)^2

Since -sin(x) = -sin(x) and cos(x)^2 = cos^2(x), the final answer is:

d/dx(sec(x)) = -sin(x)/cos(x)^2

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