Simplifying the Derivative of sec(x) | How to Find and Simplify the Derivative of sec(x) with Respect to x

d/dx [secx]

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

To find the derivative of sec(x) with respect to x, we can use the quotient rule. However, it will be easier to simplify sec(x) into its trigonometric identity before differentiating.

The trigonometric identity states that sec(x) is equal to 1/cos(x). Now, let’s differentiate sec(x) using the quotient rule.

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

According to the quotient rule, if we have a function of the form u/v, the derivative is given by:

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

In this case, u = 1 and v = cos(x). Let’s calculate the derivatives of u and v:

du/dx = 0 (since u is a constant)
dv/dx = -sin(x) (the derivative of cos(x) is -sin(x))

Plugging the derivatives into the quotient rule formula, we get:

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

Simplifying further:

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

Finally, we can simplify sin(x) / cos(x)^2 by using the trigonometric identity tan(x) = sin(x) / cos(x):

d/dx [sec(x)] = tan(x) / cos(x)^2

So, the derivative of sec(x) with respect to x is tan(x) / cos(x)^2.

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