The Quotient Rule for Differentiation | How to Find the Derivative of sec(x)

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.

First, let’s express sec(x) in terms of trigonometric functions:

sec(x) = 1/cos(x)

Now, using the quotient rule, which states that the derivative of a quotient of two functions is the numerator’s derivative times the denominator minus the denominator’s derivative times the numerator, all divided by the square of the denominator:

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

In this case, the numerator is 1 and the denominator is cos(x). Therefore:

sec'(x) = (1 * cos(x) – 0 * 1) / [cos(x)]^2
= cos(x) / [cos(x)]^2
= cos(x) / cos^2(x)
= 1 / cos(x)

Using the identity sec^2(x) = 1 + tan^2(x), we can further simplify the derivative:

sec'(x) = 1 / cos(x)
= 1 / √[1 + tan^2(x)]
= √[1 + tan^2(x)] / [1 + tan^2(x)]
= √[1 + tan^2(x)] / sec^2(x)

So, the derivative of sec(x) is √[1 + tan^2(x)] / sec^2(x).

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