The Chain Rule | Finding the Derivative of sec(x) with respect to x

d/dx[sec]

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

To find the derivative of sec(x) with respect to x, we need to use the chain rule. Let’s start by recalling the definition of secant function.

The secant function, sec(x), is defined as the reciprocal of the cosine function. In other words, sec(x) = 1/cos(x).

Now, let’s differentiate sec(x) using the chain rule. The chain rule states that if we have a composite function y = f(g(x)), then its derivative is given by dy/dx = f'(g(x)) * g'(x).

In our case, we can consider sec(x) as f(g(x)), where f(u) = 1/u and g(x) = cos(x).

Using the chain rule, the derivative of sec(x) is calculated as follows:

d/dx[sec(x)] = d/dx[1/cos(x)]
= (-1/cos^2(x)) * d/dx[cos(x)] (using the quotient rule)
= (-1/cos^2(x)) * (-sin(x)) (using the derivative of cos(x) = -sin(x))
= sin(x) / cos^2(x)

Simplifying further, we can express the derivative of sec(x) as:

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

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

More Answers: