How to Find the Derivative of Sec(x) Using Quotient Rule – Math Tutorial

(d/dx) sec(x)

sec(x)tan(x)

To find the derivative of sec(x) with respect to x, we can use the quotient rule.

Recall that sec(x) is the reciprocal of cosine (i.e., sec(x) = 1/cos(x)).

Using the quotient rule, we have:

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

The derivative of a constant (in this case, 1) is zero, so the first term simplifies to zero.

The derivative of cos(x) is -sin(x), so we have:

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

We can simplify this expression by using the identity sin^2(x) + cos^2(x) = 1 and rearranging to get:

sin^2(x) = 1 – cos^2(x)

Dividing both sides by cos^2(x), we get:

sin^2(x) / cos^2(x) = 1/cos^2(x) – 1

And since sec(x) = 1/cos(x), we can substitute to get:

sin^2(x) / cos^2(x) = sec^2(x) – 1

Substituting this into our original expression, we get:

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

So the derivative of sec(x) with respect to x is -sin(x) * sec(x) * tan(x).

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