How to Find the Derivative of sec(x): Step-by-Step Guide and Formula

ddx 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 derivative of sec(x) is given by:

d/dx(sec(x)) = sec(x) * tan(x)

Here’s a step-by-step explanation of how we get this result:

First, we can rewrite sec(x) as 1/cos(x).

d/dx(sec(x)) = d/dx(1/cos(x))

Next, we can use the quotient rule to differentiate 1/cos(x). The quotient rule states that for a function u(x)/v(x), the derivative is given by:

d/dx(u(x)/v(x)) = (v(x) * du(x)/dx – u(x) * dv(x)/dx) / (v(x))^2

In our case, u(x) = 1 and v(x) = cos(x). So let’s differentiate 1 and cos(x) separately:

du(x)/dx = 0 (since 1 is a constant)

dv(x)/dx = -sin(x) (since the derivative of cos(x) is -sin(x))

By substituting u(x), v(x), du(x)/dx, and dv(x)/dx into the quotient rule formula, we get:

d/dx(1/cos(x)) = (cos(x) * 0 – 1 * (-sin(x))) / (cos(x))^2
= sin(x) / (cos(x))^2

Now, we can simplify sin(x) / (cos(x))^2 using trigonometric identities. Since sin(x)/cos(x) is equal to tan(x), we have:

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

So, the derivative of sec(x) is sec(x) * tan(x).

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