A Comprehensive Guide on Finding the Derivative of sec(x) using the Chain Rule and Trigonometric Identities

derivative of secx

To find the derivative of sec(x), we can use the chain rule

To find the derivative of sec(x), we can use the chain rule.

The secant function, sec(x), can be defined as 1/cos(x).

Using the chain rule, we differentiate the numerator and denominator separately and then divide the two derivatives.

The derivative of 1 is 0 since it is a constant.

Now, let’s find the derivative of cos(x). The derivative of cos(x) is -sin(x).

So, the derivative of sec(x) = (0 * cos(x) – 1 * -sin(x))/(cos(x))^2

Simplifying the expression, we have:
derivative of sec(x) = sin(x)/(cos(x))^2 = sin(x)/cos^2(x)

Alternatively, we can also express the derivative of sec(x) using trigonometric identities.

By using the identity: sec^2(x) = 1 + tan^2(x), we can rewrite sec(x) as sec(x) = sqrt(1 + tan^2(x)).

Differentiating the new expression, we have:
derivative of sec(x) = derivative of sqrt(1 + tan^2(x))

Applying the chain rule, let u = 1 + tan^2(x).
The derivative of u^1/2 with respect to u is (1/2)u^(-1/2). And the derivative of tan^2(x) with respect to x is 2tan(x)sec^2(x).

So, the derivative of sec(x) = (1/2)(1 + tan^2(x))^(-1/2) * 2tan(x)sec^2(x)

Combining the terms, we get:
derivative of sec(x) = tan(x)sec(x) / sqrt(1 + tan^2(x))

Therefore, the derivative of sec(x) can be expressed as sin(x)/cos^2(x) or tan(x)sec(x) / sqrt(1 + tan^2(x)).

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