## d/dx(secx)

### To differentiate the function f(x) = sec(x), we can use the quotient rule since sec(x) is equivalent to 1/cos(x)

To differentiate the function f(x) = sec(x), we can use the quotient rule since sec(x) is equivalent to 1/cos(x). The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = [g'(x)h(x) – g(x)h'(x)]/ [h(x)]^2

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

f'(x) = [(d/dx)(1)]cos(x) – (1)(d/dx)(cos(x))] / [cos(x)]^2

The derivative of 1 with respect to x is 0, as it is a constant. The derivative of cos(x) is -sin(x). Simplifying further, we get:

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

Now, remember that the identity sec^2(x) = 1 + tan^2(x) can be derived from the Pythagorean identity sin^2(x) + cos^2(x) = 1. Rearranging this identity, we have sec^2(x) – 1 = tan^2(x). Dividing both sides by cos^2(x), we get:

1/cos^2(x) – 1 = tan^2(x)/cos^2(x)

Simplifying further, we have sec^2(x)/cos^2(x) – 1 = sec^2(x) – cos^2(x) = tan^2(x)/cos^2(x)

Therefore, we can rewrite the derivative of f(x) = sec(x) as:

f'(x) = tan^2(x)/cos^2(x)

Alternatively, we can express the result in terms of sin(x). Using the identity tan(x) = sin(x)/cos(x), we can rewrite the derivative as follows:

f'(x) = (sin^2(x)/cos^2(x)) / cos^2(x)

Simplifying further, we get:

f'(x) = sin^2(x)/cos^4(x)

So, the derivative of f(x) = sec(x) is either tan^2(x)/cos^2(x) or sin^2(x)/cos^4(x), depending on how you prefer to express it.

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