Understanding How to Find 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 start by expressing sec(x) in terms of trigonometric functions

To find the derivative of sec(x) with respect to x, we can start by expressing sec(x) in terms of trigonometric functions.

The identity sec(x) = 1/cos(x) allows us to rewrite the function as:

sec(x) = (cos(x))^(-1)

Now, we can differentiate both sides of this equation using the chain rule:

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

Let u = cos(x), then:

(d/dx) sec(x) = (d/du) [(cos(x))^(-1)] * (du/dx)

To find (d/du) [(cos(x))^(-1)], we can use the power rule of differentiation:

(d/du) [(cos(x))^(-1)] = (-1) * (cos(x))^(-2) * [d/du](cos(x))

Now, [d/du](cos(x)) is the derivative of cos(x) with respect to u, which is -sin(x):

(d/du) [(cos(x))^(-1)] = (-1) * (cos(x))^(-2) * (-sin(x))

Simplifying this expression, we have:

(d/du) [(cos(x))^(-1)] = sin(x)/(cos(x))^2

Now, we need to evaluate du/dx. We set u = cos(x), so du/dx = [d/dx](cos(x)).

The derivative of cos(x) with respect to x is -sin(x):

du/dx = [d/dx](cos(x)) = -sin(x)

Now, we substitute the results of the derivatives into the equation:

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

Simplifying further, we get:

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

Therefore, the derivative of sec(x) with respect to x is -(sin^2(x))/(cos^2(x)).

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