A Guide to Finding the Derivative of the Secant Function Using Quotient Rule and Trig Identities

d/dx 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.

The function sec(x) is defined as the reciprocal of the cosine function, which means sec(x) = 1/cos(x). To differentiate this, let’s use the quotient rule:

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

Now, the derivative of a constant is always 0, so d/dx (1) = 0. The derivative of cos(x) is -sin(x).

Substituting these values back into the equation, we get:

d/dx (sec(x)) = 0 – (-sin(x)) / (cos(x))^2
= sin(x) / (cos(x))^2

However, there is another way to express this in terms of sec(x) itself. We know that sin^2(x) + cos^2(x) = 1. By dividing both sides of the equation by cos^2(x), we can rewrite this as:

(tan^2(x) + 1) / cos^2(x) = sec^2(x)

Now, we can substitute this into our previous equation:

d/dx (sec(x)) = sin(x) / (cos(x))^2
= sin(x) * sec^2(x)

So, the derivative of sec(x) is sin(x) * sec^2(x).

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