How to Find the Derivative of sec(x) Using the Chain Rule in Calculus

d/dx sec(x)

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

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

The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

First, let’s rewrite sec(x) in terms of cosine, since sec(x) is the reciprocal of cosine.

sec(x) = 1/cos(x)

Using the chain rule, let’s differentiate 1/cos(x) with respect to x.

Here, f(u) = 1/u, where u = cos(x).
g(x) = cos(x)

To find f'(u), we differentiate 1/u with respect to u.

f'(u) = -1/u^2

To find g'(x), we differentiate cos(x) with respect to x.

g'(x) = -sin(x)

Now, using the chain rule formula, we have:

d/dx (sec(x)) = f'(g(x)) * g'(x)

= -1/(cos(x))^2 * (-sin(x))

= sin(x) / (cos(x))^2

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

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