Derivative of sec(x) with Respect to x: Step-by-Step Calculation and Explanation

𝑑/𝑑𝑥[sec 𝑥]

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

To find the derivative of sec(x) with respect to x, we can use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then its derivative is given by the derivative of the outer function times the derivative of the inner function.

In this case, the outer function is sec(x) and the inner function is x. The derivative of sec(x) is the derivative of the outer function, and the derivative of x is 1.

Now, let’s find the derivative of sec(x):

First, let’s rewrite sec(x) using its definition:
sec(x) = 1/cos(x)

Now, apply the chain rule:
d/dx[sec(x)] = d/dx[1/cos(x)]

Differentiate the numerator and the denominator separately:

For the numerator (1), the derivative is zero since it is a constant.

For the denominator (cos(x)), we need to use the chain rule again.

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

Now, substitute the derivative of the numerator and denominator into our original equation:

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

Simplifying further:

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

This is the derivative of sec(x) with respect to x.

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