How to Find the Derivative of Sec(x) Using the Quotient Rule: Step-by-Step Guide

derivative of secx

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. The quotient rule states that for functions f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative can be calculated as follows:

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

In the case of sec(x), we can rewrite it as:

sec(x) = 1/cos(x)

Now, let’s apply the quotient rule. We have:

g(x) = 1
h(x) = cos(x)

Finding the derivatives of g(x) and h(x):

g'(x) = 0 (since it’s a constant)
h'(x) = -sin(x) (using the derivative of cos(x) which is -sin(x))

Plugging these values into the quotient rule formula:

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

Now, we can simplify the expression by using the trigonometric identity sin(x)/cos(x) = tan(x):

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

Since we started with sec(x) = 1/cos(x), and the derivative of 1/cos(x) is tan(x) / cos(x)^2, we conclude that the derivative of sec(x) with respect to x is equal to:

d(sec(x))/dx = tan(x) / cos(x)^2

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