Exploring the Derivative of sec(x) and Simplifying Using Trigonometric Identities

derivative of sec x

To find the derivative of sec(x), we can start by writing it in terms of sine and cosine

To find the derivative of sec(x), we can start by writing it in terms of sine and cosine. The secant function can be expressed as:

sec(x) = 1/cos(x)

Now, we can find the derivative using the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x)/h(x), then the derivative is given by:

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

Applying the quotient rule to the function sec(x) = 1/cos(x), we have:

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

Simplifying the numerator, we get:

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

Next, we can use trigonometric identities to simplify this expression further. The identity tan(x) = sin(x)/cos(x) tells us that:

sin(x) = tan(x) * cos(x)

Substituting this into our derivative expression, we have:

f'(x) = [tan(x) * cos(x)] / [cos(x)]^2

Now, we can cancel out the common factor of cos(x) in the numerator and denominator:

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

Finally, we can use another trigonometric identity, sec(x) = 1/cos(x), to express the derivative correctly in terms of sec(x):

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

Therefore, the derivative of sec(x) is equal to tan(x) / cos(x).

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