How to Find the Derivative of the sec(x) Function Using the Quotient Rule in Calculus

Derivative of sec(x)

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

To find the derivative of the sec(x) function, we can use the quotient rule in calculus. The sec(x) function is written as 1/cos(x), where cos(x) is the denominator.

Using the quotient rule, the derivative of sec(x) can be found as follows:

Let f(x) = 1 (numerator)
Let g(x) = cos(x) (denominator)

The quotient rule states that if we have a function h(x) given by h(x) = f(x)/g(x), then the derivative of h(x) is given by:

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

Now, let’s apply this rule to find the derivative of sec(x):

f'(x) = 0, since the derivative of a constant function is always zero.
g'(x) = -sin(x), since the derivative of cos(x) is -sin(x) (using the chain rule).

Plugging these values into the quotient rule formula, we get:

(sec(x))’ = [(0)(cos(x)) – (-sin(x))(1)] / [cos(x)]^2
= sin(x) / cos(x)^2

To simplify this expression, we can rewrite sin(x) as cos(x) multiplied by tan(x):

(sec(x))’ = (cos(x)tan(x)) / cos(x)^2
= tan(x) / cos(x)

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

Note: Another way to find the derivative of sec(x) is by using trigonometric identities. The derivative of sec(x) can also be expressed as sec(x)tan(x).

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