Understanding the Derivative of the Secant Function | Step-by-Step Explanation and Simplification

(secx)’

The notation (sec x)’ represents the derivative of the secant function with respect to x, where x is the independent variable

The notation (sec x)’ represents the derivative of the secant function with respect to x, where x is the independent variable. To find this derivative, we can use the quotient rule and the chain rule.

The secant function is defined as sec x = 1 / cos x. Therefore, we can rewrite it as a quotient: sec x = 1 / cos x.

Now, let’s find the derivative of sec x.
Using the quotient rule, the derivative of sec x with respect to x can be computed as:

(sec x)’ = ( (1 / cos x)’ * cos x – 1 * (cos x)’) / (cos x)^2

To simplify this expression, we need to find the derivatives of 1 / cos x and cos x.

First, let’s find the derivative of 1 / cos x. Using the chain rule, we have:

(1 / cos x)’ = – (1 / (cos x)^2) * (-sin x) = sin x / (cos x)^2

Next, let’s find the derivative of cos x, which is simply -sin x.

Now substituting these derivatives into the original expression:

(sec x)’ = (sin x / (cos x)^2 * cos x – 1 * (-sin x)) / (cos x)^2

Simplifying further:

(sec x)’ = (sin x * cos x + sin x) / (cos x)^2

Factoring out sin x:

(sec x)’ = sin x * (cos x + 1) / (cos x)^2

Therefore, the derivative of sec x with respect to x is sin x * (cos x + 1) / (cos x)^2

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