How to Find the Derivative of Sec(x) using the Quotient Rule and Simplify the Expression

derivative of sec(x)

To find the derivative of sec(x), we can use the quotient rule

To find the derivative of sec(x), we can use the quotient rule.

Sec(x) can be written as 1/cos(x). So, using the quotient rule, we have:

f(x) = 1 and g(x) = cos(x)

f'(x) = 0 and g'(x) = -sin(x)

Now, let’s apply the quotient rule:

(f(x) * g'(x) – g(x) * f'(x)) / (g(x))^2

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

= -sin(x) / (cos(x))^2

= -sin(x) / cos^2(x)

Now, we can simplify this expression by using the identity sin^2(x) + cos^2(x) = 1. Rearranging this equation, we get sin^2(x) = 1 – cos^2(x). Dividing both sides by cos^2(x), we have sin^2(x) / cos^2(x) = (1 – cos^2(x)) / cos^2(x).

So, the derivative of sec(x) is:

– sin(x) / cos^2(x) = -1 / cos^2(x) + 1 / cos^4(x)

or

-sec(x) * tan(x)

or

-csc(x) * cot(x).

These are all equivalent forms of the derivative of sec(x).

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